Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.9% → 96.1%

Time: 13.8s

Alternatives: 16

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: 96.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|m - n\right|\\ t_1 := \frac{m + n}{2} - M\\ \mathbf{if}\;M \leq -2.3 \cdot 10^{+112}:\\ \;\;\;\;\frac{1}{e^{\left(\ell + M \cdot M\right) - t\_0}}\\ \mathbf{elif}\;M \leq 10^{+95}:\\ \;\;\;\;\frac{1 + \left(M \cdot M\right) \cdot -0.5}{e^{\left(t\_1 \cdot t\_1 + \ell\right) - t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{M \cdot M}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- m n))) (t_1 (- (/ (+ m n) 2.0) M)))
   (if (<= M -2.3e+112)
     (/ 1.0 (exp (- (+ l (* M M)) t_0)))
     (if (<= M 1e+95)
       (/ (+ 1.0 (* (* M M) -0.5)) (exp (- (+ (* t_1 t_1) l) t_0)))
       (/ (cos M) (exp (* M M)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((m - n));
	double t_1 = ((m + n) / 2.0) - M;
	double tmp;
	if (M <= -2.3e+112) {
		tmp = 1.0 / exp(((l + (M * M)) - t_0));
	} else if (M <= 1e+95) {
		tmp = (1.0 + ((M * M) * -0.5)) / exp((((t_1 * t_1) + l) - t_0));
	} else {
		tmp = cos(M) / exp((M * M));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((m - n))
    t_1 = ((m + n) / 2.0d0) - m_1
    if (m_1 <= (-2.3d+112)) then
        tmp = 1.0d0 / exp(((l + (m_1 * m_1)) - t_0))
    else if (m_1 <= 1d+95) then
        tmp = (1.0d0 + ((m_1 * m_1) * (-0.5d0))) / exp((((t_1 * t_1) + l) - t_0))
    else
        tmp = cos(m_1) / exp((m_1 * m_1))
    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.abs((m - n));
	double t_1 = ((m + n) / 2.0) - M;
	double tmp;
	if (M <= -2.3e+112) {
		tmp = 1.0 / Math.exp(((l + (M * M)) - t_0));
	} else if (M <= 1e+95) {
		tmp = (1.0 + ((M * M) * -0.5)) / Math.exp((((t_1 * t_1) + l) - t_0));
	} else {
		tmp = Math.cos(M) / Math.exp((M * M));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((m - n))
	t_1 = ((m + n) / 2.0) - M
	tmp = 0
	if M <= -2.3e+112:
		tmp = 1.0 / math.exp(((l + (M * M)) - t_0))
	elif M <= 1e+95:
		tmp = (1.0 + ((M * M) * -0.5)) / math.exp((((t_1 * t_1) + l) - t_0))
	else:
		tmp = math.cos(M) / math.exp((M * M))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(m - n))
	t_1 = Float64(Float64(Float64(m + n) / 2.0) - M)
	tmp = 0.0
	if (M <= -2.3e+112)
		tmp = Float64(1.0 / exp(Float64(Float64(l + Float64(M * M)) - t_0)));
	elseif (M <= 1e+95)
		tmp = Float64(Float64(1.0 + Float64(Float64(M * M) * -0.5)) / exp(Float64(Float64(Float64(t_1 * t_1) + l) - t_0)));
	else
		tmp = Float64(cos(M) / exp(Float64(M * M)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((m - n));
	t_1 = ((m + n) / 2.0) - M;
	tmp = 0.0;
	if (M <= -2.3e+112)
		tmp = 1.0 / exp(((l + (M * M)) - t_0));
	elseif (M <= 1e+95)
		tmp = (1.0 + ((M * M) * -0.5)) / exp((((t_1 * t_1) + l) - t_0));
	else
		tmp = cos(M) / exp((M * M));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]}, If[LessEqual[M, -2.3e+112], N[(1.0 / N[Exp[N[(N[(l + N[(M * M), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 1e+95], N[(N[(1.0 + N[(N[(M * M), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / N[Exp[N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] + l), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(M * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if M < -2.3e112

   1. Initial program 87.9%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 87.9%

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

   5. Taylor expanded in M around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 87.9%

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

   7. Simplified 87.9%

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

   8. Taylor expanded in n around inf

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 97.0%

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

   10. Simplified 97.0%

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

   11. Taylor expanded in K around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(M, M\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 100.0%

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

   if -2.3e112 < M < 1.00000000000000002e95

   1. Initial program 66.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 66.6%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 92.1%

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

   7. Simplified 92.1%

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

   8. Taylor expanded in M around 0

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

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

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left({M}^{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \ell\right), \color{blue}{\mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)}\right)\right)\right) \]

      3. unpow2 N/A

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

      4. *-lowering-*.f64 92.7%

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

   10. Simplified 92.7%

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

   if 1.00000000000000002e95 < M

   1. Initial program 80.0%

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

      1. 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 80.0%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 100.0%

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

   7. Simplified 100.0%

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

   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 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2.3 \cdot 10^{+112}:\\ \;\;\;\;\frac{1}{e^{\left(\ell + M \cdot M\right) - \left|m - n\right|}}\\ \mathbf{elif}\;M \leq 10^{+95}:\\ \;\;\;\;\frac{1 + \left(M \cdot M\right) \cdot -0.5}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{M \cdot M}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.9% accurate, 1.3× speedup

Math

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

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(Float64(t_0 * t_0) + 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[(N[(t$95$0 * t$95$0), $MachinePrecision] + l), $MachinePrecision] - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 71.5%

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

   1. 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 71.5%

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

5. Taylor expanded in K around 0

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

   1. cos-neg N/A

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

   2. cos-lowering-cos.f64 94.4%

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

7. Simplified 94.4%

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

Alternative 3: 95.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\cos M}{e^{M \cdot M}}\\ \mathbf{if}\;M \leq -1 \cdot 10^{+18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 2.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{e^{\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|m - n\right|\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (/ (cos M) (exp (* M M)))))
   (if (<= M -1e+18)
     t_0
     (if (<= M 2.5e+16)
       (/ 1.0 (exp (+ l (- (* 0.25 (* (+ m n) (+ m n))) (fabs (- m n))))))
       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 <= -1e+18) {
		tmp = t_0;
	} else if (M <= 2.5e+16) {
		tmp = 1.0 / exp((l + ((0.25 * ((m + n) * (m + n))) - fabs((m - n)))));
	} 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 <= (-1d+18)) then
        tmp = t_0
    else if (m_1 <= 2.5d+16) then
        tmp = 1.0d0 / exp((l + ((0.25d0 * ((m + n) * (m + n))) - abs((m - n)))))
    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 <= -1e+18) {
		tmp = t_0;
	} else if (M <= 2.5e+16) {
		tmp = 1.0 / Math.exp((l + ((0.25 * ((m + n) * (m + n))) - Math.abs((m - n)))));
	} 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 <= -1e+18:
		tmp = t_0
	elif M <= 2.5e+16:
		tmp = 1.0 / math.exp((l + ((0.25 * ((m + n) * (m + n))) - math.fabs((m - n)))))
	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 <= -1e+18)
		tmp = t_0;
	elseif (M <= 2.5e+16)
		tmp = Float64(1.0 / exp(Float64(l + Float64(Float64(0.25 * Float64(Float64(m + n) * Float64(m + n))) - abs(Float64(m - n))))));
	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 <= -1e+18)
		tmp = t_0;
	elseif (M <= 2.5e+16)
		tmp = 1.0 / exp((l + ((0.25 * ((m + n) * (m + n))) - abs((m - n)))));
	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, -1e+18], t$95$0, If[LessEqual[M, 2.5e+16], N[(1.0 / N[Exp[N[(l + N[(N[(0.25 * N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -1e18 or 2.5e16 < M

   1. Initial program 77.5%

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

      1. 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 77.5%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 99.1%

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

   7. Simplified 99.1%

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

   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 97.3%

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

   10. Simplified 97.3%

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

   if -1e18 < M < 2.5e16

   1. Initial program 66.9%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 66.9%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 90.7%

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

   7. Simplified 90.7%

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

   8. Taylor expanded in M around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|n - m\right|\right)\right)\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\ell + \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2}\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

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

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(m + n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right)\right) \]

      14. fabs-lowering-fabs.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right)\right) \]

      17. --lowering--.f64 90.7%

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

   10. Simplified 90.7%

       \[\leadsto \color{blue}{\frac{1}{e^{\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|n - m\right|\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1 \cdot 10^{+18}:\\ \;\;\;\;\frac{\cos M}{e^{M \cdot M}}\\ \mathbf{elif}\;M \leq 2.5 \cdot 10^{+16}:\\ \;\;\;\;\frac{1}{e^{\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|m - n\right|\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{M \cdot M}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|m - n\right|\\ \mathbf{if}\;n \leq 4.9 \cdot 10^{-104}:\\ \;\;\;\;\frac{1}{e^{\left(\ell + 0.25 \cdot \left(m \cdot m\right)\right) - t\_0}}\\ \mathbf{elif}\;n \leq 1900000:\\ \;\;\;\;\frac{1}{e^{\left(\ell + M \cdot M\right) - t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- m n))))
   (if (<= n 4.9e-104)
     (/ 1.0 (exp (- (+ l (* 0.25 (* m m))) t_0)))
     (if (<= n 1900000.0)
       (/ 1.0 (exp (- (+ l (* M M)) t_0)))
       (/ (cos M) (exp (* 0.25 (* n n))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((m - n));
	double tmp;
	if (n <= 4.9e-104) {
		tmp = 1.0 / exp(((l + (0.25 * (m * m))) - t_0));
	} else if (n <= 1900000.0) {
		tmp = 1.0 / exp(((l + (M * M)) - t_0));
	} 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) :: t_0
    real(8) :: tmp
    t_0 = abs((m - n))
    if (n <= 4.9d-104) then
        tmp = 1.0d0 / exp(((l + (0.25d0 * (m * m))) - t_0))
    else if (n <= 1900000.0d0) then
        tmp = 1.0d0 / exp(((l + (m_1 * m_1)) - t_0))
    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 t_0 = Math.abs((m - n));
	double tmp;
	if (n <= 4.9e-104) {
		tmp = 1.0 / Math.exp(((l + (0.25 * (m * m))) - t_0));
	} else if (n <= 1900000.0) {
		tmp = 1.0 / Math.exp(((l + (M * M)) - t_0));
	} else {
		tmp = Math.cos(M) / Math.exp((0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((m - n))
	tmp = 0
	if n <= 4.9e-104:
		tmp = 1.0 / math.exp(((l + (0.25 * (m * m))) - t_0))
	elif n <= 1900000.0:
		tmp = 1.0 / math.exp(((l + (M * M)) - t_0))
	else:
		tmp = math.cos(M) / math.exp((0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(m - n))
	tmp = 0.0
	if (n <= 4.9e-104)
		tmp = Float64(1.0 / exp(Float64(Float64(l + Float64(0.25 * Float64(m * m))) - t_0)));
	elseif (n <= 1900000.0)
		tmp = Float64(1.0 / exp(Float64(Float64(l + Float64(M * M)) - t_0)));
	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)
	t_0 = abs((m - n));
	tmp = 0.0;
	if (n <= 4.9e-104)
		tmp = 1.0 / exp(((l + (0.25 * (m * m))) - t_0));
	elseif (n <= 1900000.0)
		tmp = 1.0 / exp(((l + (M * M)) - t_0));
	else
		tmp = cos(M) / exp((0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, 4.9e-104], N[(1.0 / N[Exp[N[(N[(l + N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1900000.0], N[(1.0 / N[Exp[N[(N[(l + N[(M * M), $MachinePrecision]), $MachinePrecision] - t$95$0), $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 n < 4.9000000000000003e-104

   1. Initial program 74.2%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 74.2%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 93.6%

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

   7. Simplified 93.6%

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

   8. Taylor expanded in M around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|n - m\right|\right)\right)\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\ell + \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2}\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

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

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(m + n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right)\right) \]

      14. fabs-lowering-fabs.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right)\right) \]

      17. --lowering--.f64 81.0%

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

   10. Simplified 81.0%

       \[\leadsto \color{blue}{\frac{1}{e^{\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|n - m\right|\right)}}} \]

   11. Taylor expanded in n around 0

       \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{\left(\ell + \frac{1}{4} \cdot {m}^{2}\right) - \left|n - m\right|}\right)}\right) \]
   12. Step-by-step derivation

       1. fabs-sub N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(e^{\left(\ell + \frac{1}{4} \cdot {m}^{2}\right) - \left|m - n\right|}\right)\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(e^{\left(\ell + \frac{1}{4} \cdot {m}^{2}\right) - \left|m + \left(\mathsf{neg}\left(n\right)\right)\right|}\right)\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(e^{\left(\ell + \frac{1}{4} \cdot {m}^{2}\right) - \left|m + -1 \cdot n\right|}\right)\right) \]

       4. fabs-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(e^{\left(\ell + \frac{1}{4} \cdot {m}^{2}\right) - \left|\mathsf{neg}\left(\left(m + -1 \cdot n\right)\right)\right|}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {m}^{2}\right) - \left|\mathsf{neg}\left(\left(m + -1 \cdot n\right)\right)\right|\right)\right)\right) \]

       6. fabs-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {m}^{2}\right) - \left|m + -1 \cdot n\right|\right)\right)\right) \]

       7. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {m}^{2}\right) - \left|m + \left(\mathsf{neg}\left(n\right)\right)\right|\right)\right)\right) \]

       8. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {m}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

       9. fabs-sub N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {m}^{2}\right) - \left|n - m\right|\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\ell + \frac{1}{4} \cdot {m}^{2}\right), \left(\left|n - m\right|\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {m}^{2}\right)\right), \left(\left|n - m\right|\right)\right)\right)\right) \]

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

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

       13. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{4}, \left(m \cdot m\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right) \]

       15. sub-neg N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right) \]

       16. mul-1-neg N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right) \]

       17. fabs-lowering-fabs.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right) \]

       18. mul-1-neg N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right) \]

       19. sub-neg N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right) \]

       20. --lowering--.f64 59.6%

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

   13. Simplified 59.6%

       \[\leadsto \frac{1}{\color{blue}{e^{\left(\ell + 0.25 \cdot \left(m \cdot m\right)\right) - \left|n - m\right|}}} \]

   if 4.9000000000000003e-104 < n < 1.9e6

   1. Initial program 66.9%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 66.9%

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

   5. Taylor expanded in M around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 59.4%

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

   7. Simplified 59.4%

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

   8. Taylor expanded in n around inf

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 69.6%

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

   10. Simplified 69.6%

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

   11. Taylor expanded in K around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(M, M\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 69.6%

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

   if 1.9e6 < n

   1. Initial program 63.8%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 63.8%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 100.0%

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

   7. Simplified 100.0%

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

   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 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 4.9 \cdot 10^{-104}:\\ \;\;\;\;\frac{1}{e^{\left(\ell + 0.25 \cdot \left(m \cdot m\right)\right) - \left|m - n\right|}}\\ \mathbf{elif}\;n \leq 1900000:\\ \;\;\;\;\frac{1}{e^{\left(\ell + M \cdot M\right) - \left|m - n\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 4.2 \cdot 10^{-213}:\\ \;\;\;\;\frac{\cos M}{e^{0.25 \cdot \left(m \cdot m\right)}}\\ \mathbf{elif}\;n \leq 1900000:\\ \;\;\;\;\frac{1}{e^{\left(\ell + M \cdot M\right) - \left|m - n\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 4.2e-213)
   (/ (cos M) (exp (* 0.25 (* m m))))
   (if (<= n 1900000.0)
     (/ 1.0 (exp (- (+ l (* M M)) (fabs (- m n)))))
     (/ (cos M) (exp (* 0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 4.2e-213) {
		tmp = cos(M) / exp((0.25 * (m * m)));
	} else if (n <= 1900000.0) {
		tmp = 1.0 / exp(((l + (M * M)) - fabs((m - n))));
	} 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 (n <= 4.2d-213) then
        tmp = cos(m_1) / exp((0.25d0 * (m * m)))
    else if (n <= 1900000.0d0) then
        tmp = 1.0d0 / exp(((l + (m_1 * m_1)) - abs((m - n))))
    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 (n <= 4.2e-213) {
		tmp = Math.cos(M) / Math.exp((0.25 * (m * m)));
	} else if (n <= 1900000.0) {
		tmp = 1.0 / Math.exp(((l + (M * M)) - Math.abs((m - n))));
	} else {
		tmp = Math.cos(M) / Math.exp((0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 4.2e-213:
		tmp = math.cos(M) / math.exp((0.25 * (m * m)))
	elif n <= 1900000.0:
		tmp = 1.0 / math.exp(((l + (M * M)) - math.fabs((m - n))))
	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 (n <= 4.2e-213)
		tmp = Float64(cos(M) / exp(Float64(0.25 * Float64(m * m))));
	elseif (n <= 1900000.0)
		tmp = Float64(1.0 / exp(Float64(Float64(l + Float64(M * M)) - abs(Float64(m - n)))));
	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 (n <= 4.2e-213)
		tmp = cos(M) / exp((0.25 * (m * m)));
	elseif (n <= 1900000.0)
		tmp = 1.0 / exp(((l + (M * M)) - abs((m - n))));
	else
		tmp = cos(M) / exp((0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 4.2e-213], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1900000.0], N[(1.0 / N[Exp[N[(N[(l + N[(M * M), $MachinePrecision]), $MachinePrecision] - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $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 n < 4.1999999999999997e-213

   1. Initial program 72.8%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 72.8%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 92.6%

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

   7. Simplified 92.6%

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

   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 52.6%

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

   10. Simplified 52.6%

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

   if 4.1999999999999997e-213 < n < 1.9e6

   1. Initial program 74.2%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 74.2%

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

   5. Taylor expanded in M around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 60.5%

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

   7. Simplified 60.5%

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

   8. Taylor expanded in n around inf

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 70.0%

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

   10. Simplified 70.0%

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

   11. Taylor expanded in K around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(M, M\right), \ell\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 70.0%

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

   if 1.9e6 < n

   1. Initial program 63.8%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 63.8%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 100.0%

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

   7. Simplified 100.0%

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

   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 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 4.2 \cdot 10^{-213}:\\ \;\;\;\;\frac{\cos M}{e^{0.25 \cdot \left(m \cdot m\right)}}\\ \mathbf{elif}\;n \leq 1900000:\\ \;\;\;\;\frac{1}{e^{\left(\ell + M \cdot M\right) - \left|m - n\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -54:\\ \;\;\;\;\frac{1}{e^{0.25 \cdot \left(m \cdot m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{\ell + \left(0.25 \cdot \left(n \cdot n\right) - \left|m - n\right|\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -54.0)
   (/ 1.0 (exp (* 0.25 (* m m))))
   (/ 1.0 (exp (+ l (- (* 0.25 (* n n)) (fabs (- m n))))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -54.0) {
		tmp = 1.0 / exp((0.25 * (m * m)));
	} else {
		tmp = 1.0 / exp((l + ((0.25 * (n * n)) - fabs((m - n)))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-54.0d0)) then
        tmp = 1.0d0 / exp((0.25d0 * (m * m)))
    else
        tmp = 1.0d0 / exp((l + ((0.25d0 * (n * n)) - abs((m - n)))))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -54.0], N[(1.0 / N[Exp[N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Exp[N[(l + N[(N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision] - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -54

   1. Initial program 66.2%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 66.2%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in M around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|n - m\right|\right)\right)\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\ell + \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2}\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

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

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(m + n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right)\right) \]

      14. fabs-lowering-fabs.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right)\right) \]

      17. --lowering--.f64 93.9%

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

   10. Simplified 93.9%

       \[\leadsto \color{blue}{\frac{1}{e^{\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|n - m\right|\right)}}} \]

   11. Taylor expanded in m around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 95.5%

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

   13. Simplified 95.5%

       \[\leadsto \frac{1}{e^{\color{blue}{0.25 \cdot \left(m \cdot m\right)}}} \]

   if -54 < m

   1. Initial program 73.3%

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

      1. 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 73.3%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 93.0%

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

   7. Simplified 93.0%

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

   8. Taylor expanded in M around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|n - m\right|\right)\right)\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\ell + \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2}\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

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

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(m + n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right)\right) \]

      14. fabs-lowering-fabs.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right)\right) \]

      17. --lowering--.f64 79.2%

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

   10. Simplified 79.2%

       \[\leadsto \color{blue}{\frac{1}{e^{\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|n - m\right|\right)}}} \]

   11. Taylor expanded in m around 0

       \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{\left(\ell + \frac{1}{4} \cdot {n}^{2}\right) - \left|n - m\right|}\right)}\right) \]
   12. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(e^{\left(\ell + \frac{1}{4} \cdot {n}^{2}\right) - \left|n + \left(\mathsf{neg}\left(m\right)\right)\right|}\right)\right) \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(e^{\left(\ell + \frac{1}{4} \cdot {n}^{2}\right) - \left|n + -1 \cdot m\right|}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {n}^{2}\right) - \left|n + -1 \cdot m\right|\right)\right)\right) \]

       4. associate--l+ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\ell + \left(\frac{1}{4} \cdot {n}^{2} - \left|n + -1 \cdot m\right|\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {n}^{2} - \left|n + -1 \cdot m\right|\right)\right)\right)\right) \]

       6. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {n}^{2} - \left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right) \]

       7. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {n}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\left(\frac{1}{4} \cdot {n}^{2}\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

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

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(n \cdot n\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(n, n\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

       12. sub-neg N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(n, n\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right)\right) \]

       13. mul-1-neg N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(n, n\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right)\right) \]

       14. fabs-lowering-fabs.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(n, n\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right)\right) \]

       15. mul-1-neg N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(n, n\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right)\right) \]

       16. sub-neg N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(n, n\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right)\right) \]

       17. --lowering--.f64 60.5%

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

   13. Simplified 60.5%

       \[\leadsto \frac{1}{\color{blue}{e^{\ell + \left(0.25 \cdot \left(n \cdot n\right) - \left|n - m\right|\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.4%

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

Alternative 7: 63.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 5.6 \cdot 10^{-102}:\\ \;\;\;\;\frac{1}{e^{0.25 \cdot \left(m \cdot m\right) - \left|m - n\right|}}\\ \mathbf{elif}\;n \leq 94000000000000:\\ \;\;\;\;\frac{\cos M}{e^{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 5.6e-102)
   (/ 1.0 (exp (- (* 0.25 (* m m)) (fabs (- m n)))))
   (if (<= n 94000000000000.0)
     (/ (cos M) (exp (* M M)))
     (/ 1.0 (exp (* 0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 5.6e-102) {
		tmp = 1.0 / exp(((0.25 * (m * m)) - fabs((m - n))));
	} else if (n <= 94000000000000.0) {
		tmp = cos(M) / exp((M * M));
	} else {
		tmp = 1.0 / 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 <= 5.6d-102) then
        tmp = 1.0d0 / exp(((0.25d0 * (m * m)) - abs((m - n))))
    else if (n <= 94000000000000.0d0) then
        tmp = cos(m_1) / exp((m_1 * m_1))
    else
        tmp = 1.0d0 / 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 <= 5.6e-102) {
		tmp = 1.0 / Math.exp(((0.25 * (m * m)) - Math.abs((m - n))));
	} else if (n <= 94000000000000.0) {
		tmp = Math.cos(M) / Math.exp((M * M));
	} else {
		tmp = 1.0 / Math.exp((0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 5.6e-102:
		tmp = 1.0 / math.exp(((0.25 * (m * m)) - math.fabs((m - n))))
	elif n <= 94000000000000.0:
		tmp = math.cos(M) / math.exp((M * M))
	else:
		tmp = 1.0 / math.exp((0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 5.6e-102)
		tmp = Float64(1.0 / exp(Float64(Float64(0.25 * Float64(m * m)) - abs(Float64(m - n)))));
	elseif (n <= 94000000000000.0)
		tmp = Float64(cos(M) / exp(Float64(M * M)));
	else
		tmp = Float64(1.0 / 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 <= 5.6e-102)
		tmp = 1.0 / exp(((0.25 * (m * m)) - abs((m - n))));
	elseif (n <= 94000000000000.0)
		tmp = cos(M) / exp((M * M));
	else
		tmp = 1.0 / exp((0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 5.6e-102], N[(1.0 / N[Exp[N[(N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision] - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 94000000000000.0], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(M * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Exp[N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 5.60000000000000026e-102

   1. Initial program 74.2%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 74.2%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 93.6%

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

   7. Simplified 93.6%

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

   8. Taylor expanded in M around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|n - m\right|\right)\right)\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\ell + \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2}\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

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

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(m + n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right)\right) \]

      14. fabs-lowering-fabs.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right)\right) \]

      17. --lowering--.f64 81.0%

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

   10. Simplified 81.0%

       \[\leadsto \color{blue}{\frac{1}{e^{\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|n - m\right|\right)}}} \]

   11. Taylor expanded in n around 0

       \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{\left(\ell + \frac{1}{4} \cdot {m}^{2}\right) - \left|n - m\right|}\right)}\right) \]
   12. Step-by-step derivation

       1. fabs-sub N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(e^{\left(\ell + \frac{1}{4} \cdot {m}^{2}\right) - \left|m - n\right|}\right)\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(e^{\left(\ell + \frac{1}{4} \cdot {m}^{2}\right) - \left|m + \left(\mathsf{neg}\left(n\right)\right)\right|}\right)\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(e^{\left(\ell + \frac{1}{4} \cdot {m}^{2}\right) - \left|m + -1 \cdot n\right|}\right)\right) \]

       4. fabs-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(e^{\left(\ell + \frac{1}{4} \cdot {m}^{2}\right) - \left|\mathsf{neg}\left(\left(m + -1 \cdot n\right)\right)\right|}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {m}^{2}\right) - \left|\mathsf{neg}\left(\left(m + -1 \cdot n\right)\right)\right|\right)\right)\right) \]

       6. fabs-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {m}^{2}\right) - \left|m + -1 \cdot n\right|\right)\right)\right) \]

       7. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {m}^{2}\right) - \left|m + \left(\mathsf{neg}\left(n\right)\right)\right|\right)\right)\right) \]

       8. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {m}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

       9. fabs-sub N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {m}^{2}\right) - \left|n - m\right|\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\ell + \frac{1}{4} \cdot {m}^{2}\right), \left(\left|n - m\right|\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {m}^{2}\right)\right), \left(\left|n - m\right|\right)\right)\right)\right) \]

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

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

       13. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{4}, \left(m \cdot m\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right) \]

       15. sub-neg N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right) \]

       16. mul-1-neg N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right) \]

       17. fabs-lowering-fabs.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right) \]

       18. mul-1-neg N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right) \]

       19. sub-neg N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right) \]

       20. --lowering--.f64 59.6%

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

   13. Simplified 59.6%

       \[\leadsto \frac{1}{\color{blue}{e^{\left(\ell + 0.25 \cdot \left(m \cdot m\right)\right) - \left|n - m\right|}}} \]

   14. Taylor expanded in l around 0

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 48.8%

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

   16. Simplified 48.8%

       \[\leadsto \frac{1}{e^{\color{blue}{0.25 \cdot \left(m \cdot m\right)} - \left|n - m\right|}} \]

   if 5.60000000000000026e-102 < n < 9.4e13

   1. Initial program 66.9%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 66.9%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 91.1%

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

   7. Simplified 91.1%

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

   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 60.0%

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

   10. Simplified 60.0%

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

   if 9.4e13 < n

   1. Initial program 63.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 63.6%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in M around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|n - m\right|\right)\right)\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\ell + \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2}\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

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

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(m + n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right)\right) \]

      14. fabs-lowering-fabs.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right)\right) \]

      17. --lowering--.f64 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{1}{e^{\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|n - m\right|\right)}}} \]

   11. Taylor expanded in n around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 100.0%

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

   13. Simplified 100.0%

       \[\leadsto \frac{1}{e^{\color{blue}{0.25 \cdot \left(n \cdot n\right)}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 5.6 \cdot 10^{-102}:\\ \;\;\;\;\frac{1}{e^{0.25 \cdot \left(m \cdot m\right) - \left|m - n\right|}}\\ \mathbf{elif}\;n \leq 94000000000000:\\ \;\;\;\;\frac{\cos M}{e^{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 5.2 \cdot 10^{-210}:\\ \;\;\;\;\frac{\cos M}{e^{0.25 \cdot \left(m \cdot m\right)}}\\ \mathbf{elif}\;n \leq 94000000000000:\\ \;\;\;\;\frac{\cos M}{e^{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 5.2e-210)
   (/ (cos M) (exp (* 0.25 (* m m))))
   (if (<= n 94000000000000.0)
     (/ (cos M) (exp (* M M)))
     (/ 1.0 (exp (* 0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 5.2e-210) {
		tmp = cos(M) / exp((0.25 * (m * m)));
	} else if (n <= 94000000000000.0) {
		tmp = cos(M) / exp((M * M));
	} else {
		tmp = 1.0 / 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 <= 5.2d-210) then
        tmp = cos(m_1) / exp((0.25d0 * (m * m)))
    else if (n <= 94000000000000.0d0) then
        tmp = cos(m_1) / exp((m_1 * m_1))
    else
        tmp = 1.0d0 / 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 <= 5.2e-210) {
		tmp = Math.cos(M) / Math.exp((0.25 * (m * m)));
	} else if (n <= 94000000000000.0) {
		tmp = Math.cos(M) / Math.exp((M * M));
	} else {
		tmp = 1.0 / Math.exp((0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 5.2e-210:
		tmp = math.cos(M) / math.exp((0.25 * (m * m)))
	elif n <= 94000000000000.0:
		tmp = math.cos(M) / math.exp((M * M))
	else:
		tmp = 1.0 / math.exp((0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 5.2e-210)
		tmp = Float64(cos(M) / exp(Float64(0.25 * Float64(m * m))));
	elseif (n <= 94000000000000.0)
		tmp = Float64(cos(M) / exp(Float64(M * M)));
	else
		tmp = Float64(1.0 / 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 <= 5.2e-210)
		tmp = cos(M) / exp((0.25 * (m * m)));
	elseif (n <= 94000000000000.0)
		tmp = cos(M) / exp((M * M));
	else
		tmp = 1.0 / exp((0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 5.2e-210], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 94000000000000.0], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(M * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Exp[N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 5.1999999999999997e-210

   1. Initial program 72.8%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 72.8%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 92.6%

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

   7. Simplified 92.6%

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

   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 52.6%

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

   10. Simplified 52.6%

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

   if 5.1999999999999997e-210 < n < 9.4e13

   1. Initial program 73.8%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 73.8%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 94.9%

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

   7. Simplified 94.9%

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

   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 64.0%

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

   10. Simplified 64.0%

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

   if 9.4e13 < n

   1. Initial program 63.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 63.6%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in M around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|n - m\right|\right)\right)\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\ell + \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2}\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

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

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(m + n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right)\right) \]

      14. fabs-lowering-fabs.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right)\right) \]

      17. --lowering--.f64 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{1}{e^{\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|n - m\right|\right)}}} \]

   11. Taylor expanded in n around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 100.0%

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

   13. Simplified 100.0%

       \[\leadsto \frac{1}{e^{\color{blue}{0.25 \cdot \left(n \cdot n\right)}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 65.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 10^{-208}:\\ \;\;\;\;\frac{1}{e^{0.25 \cdot \left(m \cdot m\right)}}\\ \mathbf{elif}\;n \leq 94000000000000:\\ \;\;\;\;\frac{\cos M}{e^{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 1e-208)
   (/ 1.0 (exp (* 0.25 (* m m))))
   (if (<= n 94000000000000.0)
     (/ (cos M) (exp (* M M)))
     (/ 1.0 (exp (* 0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1e-208) {
		tmp = 1.0 / exp((0.25 * (m * m)));
	} else if (n <= 94000000000000.0) {
		tmp = cos(M) / exp((M * M));
	} else {
		tmp = 1.0 / 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-208) then
        tmp = 1.0d0 / exp((0.25d0 * (m * m)))
    else if (n <= 94000000000000.0d0) then
        tmp = cos(m_1) / exp((m_1 * m_1))
    else
        tmp = 1.0d0 / 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-208) {
		tmp = 1.0 / Math.exp((0.25 * (m * m)));
	} else if (n <= 94000000000000.0) {
		tmp = Math.cos(M) / Math.exp((M * M));
	} else {
		tmp = 1.0 / Math.exp((0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 1e-208:
		tmp = 1.0 / math.exp((0.25 * (m * m)))
	elif n <= 94000000000000.0:
		tmp = math.cos(M) / math.exp((M * M))
	else:
		tmp = 1.0 / math.exp((0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 1e-208)
		tmp = Float64(1.0 / exp(Float64(0.25 * Float64(m * m))));
	elseif (n <= 94000000000000.0)
		tmp = Float64(cos(M) / exp(Float64(M * M)));
	else
		tmp = Float64(1.0 / 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-208)
		tmp = 1.0 / exp((0.25 * (m * m)));
	elseif (n <= 94000000000000.0)
		tmp = cos(M) / exp((M * M));
	else
		tmp = 1.0 / exp((0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 1e-208], N[(1.0 / N[Exp[N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 94000000000000.0], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(M * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Exp[N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 1.0000000000000001e-208

   1. Initial program 72.8%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 72.8%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 92.6%

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

   7. Simplified 92.6%

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

   8. Taylor expanded in M around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|n - m\right|\right)\right)\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\ell + \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2}\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

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

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(m + n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right)\right) \]

      14. fabs-lowering-fabs.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right)\right) \]

      17. --lowering--.f64 81.9%

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

   10. Simplified 81.9%

       \[\leadsto \color{blue}{\frac{1}{e^{\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|n - m\right|\right)}}} \]

   11. Taylor expanded in m around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 52.6%

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

   13. Simplified 52.6%

       \[\leadsto \frac{1}{e^{\color{blue}{0.25 \cdot \left(m \cdot m\right)}}} \]

   if 1.0000000000000001e-208 < n < 9.4e13

   1. Initial program 73.8%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 73.8%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 94.9%

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

   7. Simplified 94.9%

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

   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 64.0%

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

   10. Simplified 64.0%

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

   if 9.4e13 < n

   1. Initial program 63.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 63.6%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in M around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|n - m\right|\right)\right)\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\ell + \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2}\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

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

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(m + n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right)\right) \]

      14. fabs-lowering-fabs.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right)\right) \]

      17. --lowering--.f64 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{1}{e^{\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|n - m\right|\right)}}} \]

   11. Taylor expanded in n around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 100.0%

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

   13. Simplified 100.0%

       \[\leadsto \frac{1}{e^{\color{blue}{0.25 \cdot \left(n \cdot n\right)}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 63.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 5.5 \cdot 10^{-161}:\\ \;\;\;\;\frac{1}{e^{0.25 \cdot \left(m \cdot m\right)}}\\ \mathbf{elif}\;n \leq 94000000000000:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 5.5e-161)
   (/ 1.0 (exp (* 0.25 (* m m))))
   (if (<= n 94000000000000.0)
     (/ (cos M) (exp l))
     (/ 1.0 (exp (* 0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 5.5e-161) {
		tmp = 1.0 / exp((0.25 * (m * m)));
	} else if (n <= 94000000000000.0) {
		tmp = cos(M) / exp(l);
	} else {
		tmp = 1.0 / 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 <= 5.5d-161) then
        tmp = 1.0d0 / exp((0.25d0 * (m * m)))
    else if (n <= 94000000000000.0d0) then
        tmp = cos(m_1) / exp(l)
    else
        tmp = 1.0d0 / 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 <= 5.5e-161) {
		tmp = 1.0 / Math.exp((0.25 * (m * m)));
	} else if (n <= 94000000000000.0) {
		tmp = Math.cos(M) / Math.exp(l);
	} else {
		tmp = 1.0 / Math.exp((0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 5.5e-161:
		tmp = 1.0 / math.exp((0.25 * (m * m)))
	elif n <= 94000000000000.0:
		tmp = math.cos(M) / math.exp(l)
	else:
		tmp = 1.0 / math.exp((0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 5.5e-161)
		tmp = Float64(1.0 / exp(Float64(0.25 * Float64(m * m))));
	elseif (n <= 94000000000000.0)
		tmp = Float64(cos(M) / exp(l));
	else
		tmp = Float64(1.0 / 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 <= 5.5e-161)
		tmp = 1.0 / exp((0.25 * (m * m)));
	elseif (n <= 94000000000000.0)
		tmp = cos(M) / exp(l);
	else
		tmp = 1.0 / exp((0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 5.5e-161], N[(1.0 / N[Exp[N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 94000000000000.0], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Exp[N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 5.5e-161

   1. Initial program 73.5%

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

      1. 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 73.5%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 93.2%

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

   7. Simplified 93.2%

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

   8. Taylor expanded in M around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|n - m\right|\right)\right)\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\ell + \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2}\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

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

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(m + n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right)\right) \]

      14. fabs-lowering-fabs.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right)\right) \]

      17. --lowering--.f64 81.2%

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

   10. Simplified 81.2%

       \[\leadsto \color{blue}{\frac{1}{e^{\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|n - m\right|\right)}}} \]

   11. Taylor expanded in m around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 53.3%

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

   13. Simplified 53.3%

       \[\leadsto \frac{1}{e^{\color{blue}{0.25 \cdot \left(m \cdot m\right)}}} \]

   if 5.5e-161 < n < 9.4e13

   1. Initial program 71.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 71.6%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 93.0%

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

   7. Simplified 93.0%

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

   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 41.8%

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

   if 9.4e13 < n

   1. Initial program 63.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 63.6%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in M around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|n - m\right|\right)\right)\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\ell + \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2}\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

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

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(m + n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right)\right) \]

      14. fabs-lowering-fabs.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right)\right) \]

      17. --lowering--.f64 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{1}{e^{\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|n - m\right|\right)}}} \]

   11. Taylor expanded in n around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 100.0%

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

   13. Simplified 100.0%

       \[\leadsto \frac{1}{e^{\color{blue}{0.25 \cdot \left(n \cdot n\right)}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 64.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 3.6 \cdot 10^{-73}:\\ \;\;\;\;\frac{1}{e^{0.25 \cdot \left(m \cdot m\right)}}\\ \mathbf{elif}\;n \leq 94000000000000:\\ \;\;\;\;\frac{1 + \left(M \cdot M\right) \cdot -0.5}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 3.6e-73)
   (/ 1.0 (exp (* 0.25 (* m m))))
   (if (<= n 94000000000000.0)
     (/ (+ 1.0 (* (* M M) -0.5)) (exp l))
     (/ 1.0 (exp (* 0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 3.6e-73) {
		tmp = 1.0 / exp((0.25 * (m * m)));
	} else if (n <= 94000000000000.0) {
		tmp = (1.0 + ((M * M) * -0.5)) / exp(l);
	} else {
		tmp = 1.0 / 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 <= 3.6d-73) then
        tmp = 1.0d0 / exp((0.25d0 * (m * m)))
    else if (n <= 94000000000000.0d0) then
        tmp = (1.0d0 + ((m_1 * m_1) * (-0.5d0))) / exp(l)
    else
        tmp = 1.0d0 / 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 <= 3.6e-73) {
		tmp = 1.0 / Math.exp((0.25 * (m * m)));
	} else if (n <= 94000000000000.0) {
		tmp = (1.0 + ((M * M) * -0.5)) / Math.exp(l);
	} else {
		tmp = 1.0 / Math.exp((0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 3.6e-73:
		tmp = 1.0 / math.exp((0.25 * (m * m)))
	elif n <= 94000000000000.0:
		tmp = (1.0 + ((M * M) * -0.5)) / math.exp(l)
	else:
		tmp = 1.0 / math.exp((0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 3.6e-73)
		tmp = Float64(1.0 / exp(Float64(0.25 * Float64(m * m))));
	elseif (n <= 94000000000000.0)
		tmp = Float64(Float64(1.0 + Float64(Float64(M * M) * -0.5)) / exp(l));
	else
		tmp = Float64(1.0 / 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 <= 3.6e-73)
		tmp = 1.0 / exp((0.25 * (m * m)));
	elseif (n <= 94000000000000.0)
		tmp = (1.0 + ((M * M) * -0.5)) / exp(l);
	else
		tmp = 1.0 / exp((0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 3.6e-73], N[(1.0 / N[Exp[N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 94000000000000.0], N[(N[(1.0 + N[(N[(M * M), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Exp[N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 3.5999999999999999e-73

   1. Initial program 74.4%

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

      1. 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 74.4%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 93.8%

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

   7. Simplified 93.8%

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

   8. Taylor expanded in M around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|n - m\right|\right)\right)\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\ell + \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2}\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

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

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(m + n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right)\right) \]

      14. fabs-lowering-fabs.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right)\right) \]

      17. --lowering--.f64 80.5%

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

   10. Simplified 80.5%

       \[\leadsto \color{blue}{\frac{1}{e^{\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|n - m\right|\right)}}} \]

   11. Taylor expanded in m around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 54.0%

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

   13. Simplified 54.0%

       \[\leadsto \frac{1}{e^{\color{blue}{0.25 \cdot \left(m \cdot m\right)}}} \]

   if 3.5999999999999999e-73 < n < 9.4e13

   1. Initial program 64.5%

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

      1. 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 64.5%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 89.5%

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

   7. Simplified 89.5%

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

   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 42.3%

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

   11. Taylor expanded in M around 0

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

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

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

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

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 42.0%

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

   13. Simplified 42.0%

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

   if 9.4e13 < n

   1. Initial program 63.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 63.6%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in M around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|n - m\right|\right)\right)\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\ell + \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2}\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

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

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(m + n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right)\right) \]

      14. fabs-lowering-fabs.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right)\right) \]

      17. --lowering--.f64 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{1}{e^{\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|n - m\right|\right)}}} \]

   11. Taylor expanded in n around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 100.0%

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

   13. Simplified 100.0%

       \[\leadsto \frac{1}{e^{\color{blue}{0.25 \cdot \left(n \cdot n\right)}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 3.6 \cdot 10^{-73}:\\ \;\;\;\;\frac{1}{e^{0.25 \cdot \left(m \cdot m\right)}}\\ \mathbf{elif}\;n \leq 94000000000000:\\ \;\;\;\;\frac{1 + \left(M \cdot M\right) \cdot -0.5}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 6.4 \cdot 10^{-161}:\\ \;\;\;\;\frac{1}{e^{0.25 \cdot \left(m \cdot m\right)}}\\ \mathbf{elif}\;n \leq 94000000000000:\\ \;\;\;\;\frac{1}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 6.4e-161)
   (/ 1.0 (exp (* 0.25 (* m m))))
   (if (<= n 94000000000000.0)
     (/ 1.0 (exp l))
     (/ 1.0 (exp (* 0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 6.4e-161) {
		tmp = 1.0 / exp((0.25 * (m * m)));
	} else if (n <= 94000000000000.0) {
		tmp = 1.0 / exp(l);
	} else {
		tmp = 1.0 / 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 <= 6.4d-161) then
        tmp = 1.0d0 / exp((0.25d0 * (m * m)))
    else if (n <= 94000000000000.0d0) then
        tmp = 1.0d0 / exp(l)
    else
        tmp = 1.0d0 / 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 <= 6.4e-161) {
		tmp = 1.0 / Math.exp((0.25 * (m * m)));
	} else if (n <= 94000000000000.0) {
		tmp = 1.0 / Math.exp(l);
	} else {
		tmp = 1.0 / Math.exp((0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 6.4e-161:
		tmp = 1.0 / math.exp((0.25 * (m * m)))
	elif n <= 94000000000000.0:
		tmp = 1.0 / math.exp(l)
	else:
		tmp = 1.0 / math.exp((0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 6.4e-161)
		tmp = Float64(1.0 / exp(Float64(0.25 * Float64(m * m))));
	elseif (n <= 94000000000000.0)
		tmp = Float64(1.0 / exp(l));
	else
		tmp = Float64(1.0 / 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 <= 6.4e-161)
		tmp = 1.0 / exp((0.25 * (m * m)));
	elseif (n <= 94000000000000.0)
		tmp = 1.0 / exp(l);
	else
		tmp = 1.0 / exp((0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 6.4e-161], N[(1.0 / N[Exp[N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 94000000000000.0], N[(1.0 / N[Exp[l], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Exp[N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 6.39999999999999971e-161

   1. Initial program 73.5%

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

      1. 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 73.5%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 93.2%

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

   7. Simplified 93.2%

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

   8. Taylor expanded in M around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|n - m\right|\right)\right)\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\ell + \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2}\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

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

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(m + n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right)\right) \]

      14. fabs-lowering-fabs.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right)\right) \]

      17. --lowering--.f64 81.2%

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

   10. Simplified 81.2%

       \[\leadsto \color{blue}{\frac{1}{e^{\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|n - m\right|\right)}}} \]

   11. Taylor expanded in m around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 53.3%

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

   13. Simplified 53.3%

       \[\leadsto \frac{1}{e^{\color{blue}{0.25 \cdot \left(m \cdot m\right)}}} \]

   if 6.39999999999999971e-161 < n < 9.4e13

   1. Initial program 71.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 71.6%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 93.0%

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

   7. Simplified 93.0%

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

   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 41.8%

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

   11. Taylor expanded in M around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{exp.f64}\left(\ell\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 41.8%

       \[\leadsto \frac{\color{blue}{1}}{e^{\ell}} \]

   if 9.4e13 < n

   1. Initial program 63.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 63.6%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in M around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|n - m\right|\right)\right)\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\ell + \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2}\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

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

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(m + n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right)\right) \]

      14. fabs-lowering-fabs.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right)\right) \]

      17. --lowering--.f64 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{1}{e^{\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|n - m\right|\right)}}} \]

   11. Taylor expanded in n around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 100.0%

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

   13. Simplified 100.0%

       \[\leadsto \frac{1}{e^{\color{blue}{0.25 \cdot \left(n \cdot n\right)}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 69.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{e^{0.25 \cdot \left(m \cdot m\right)}}\\ \mathbf{if}\;m \leq -54:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 3.3 \cdot 10^{-12}:\\ \;\;\;\;\frac{1}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (/ 1.0 (exp (* 0.25 (* m m))))))
   (if (<= m -54.0) t_0 (if (<= m 3.3e-12) (/ 1.0 (exp l)) t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = 1.0 / exp((0.25 * (m * m)));
	double tmp;
	if (m <= -54.0) {
		tmp = t_0;
	} else if (m <= 3.3e-12) {
		tmp = 1.0 / exp(l);
	} 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 = 1.0d0 / exp((0.25d0 * (m * m)))
    if (m <= (-54.0d0)) then
        tmp = t_0
    else if (m <= 3.3d-12) then
        tmp = 1.0d0 / exp(l)
    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 = 1.0 / Math.exp((0.25 * (m * m)));
	double tmp;
	if (m <= -54.0) {
		tmp = t_0;
	} else if (m <= 3.3e-12) {
		tmp = 1.0 / Math.exp(l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = 1.0 / math.exp((0.25 * (m * m)))
	tmp = 0
	if m <= -54.0:
		tmp = t_0
	elif m <= 3.3e-12:
		tmp = 1.0 / math.exp(l)
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(1.0 / exp(Float64(0.25 * Float64(m * m))))
	tmp = 0.0
	if (m <= -54.0)
		tmp = t_0;
	elseif (m <= 3.3e-12)
		tmp = Float64(1.0 / exp(l));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = 1.0 / exp((0.25 * (m * m)));
	tmp = 0.0;
	if (m <= -54.0)
		tmp = t_0;
	elseif (m <= 3.3e-12)
		tmp = 1.0 / exp(l);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(1.0 / N[Exp[N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -54.0], t$95$0, If[LessEqual[m, 3.3e-12], N[(1.0 / N[Exp[l], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -54 or 3.3000000000000001e-12 < m

   1. Initial program 67.7%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 67.7%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 99.2%

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

   7. Simplified 99.2%

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

   8. Taylor expanded in M around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

      3. fabs-sub N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|n - m\right|\right)\right)\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\ell + \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|n - m\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2}\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

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

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(m + n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \left(m + n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right)\right) \]

      14. fabs-lowering-fabs.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right)\right) \]

      17. --lowering--.f64 94.8%

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

   10. Simplified 94.8%

       \[\leadsto \color{blue}{\frac{1}{e^{\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|n - m\right|\right)}}} \]

   11. Taylor expanded in m around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 93.4%

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

   13. Simplified 93.4%

       \[\leadsto \frac{1}{e^{\color{blue}{0.25 \cdot \left(m \cdot m\right)}}} \]

   if -54 < m < 3.3000000000000001e-12

   1. Initial program 75.6%

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

      1. 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 75.6%

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

   5. Taylor expanded in K around 0

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

      1. cos-neg N/A

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

      2. cos-lowering-cos.f64 89.1%

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

   7. Simplified 89.1%

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

   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 40.8%

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

   11. Taylor expanded in M around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{exp.f64}\left(\ell\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 40.8%

       \[\leadsto \frac{\color{blue}{1}}{e^{\ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 35.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{e^{\ell}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(K, m, n, M, l):
	return 1.0 / math.exp(l)

Julia

function code(K, m, n, M, l)
	return Float64(1.0 / exp(l))
end

MATLAB

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

Wolfram

code[K_, m_, n_, M_, l_] := N[(1.0 / N[Exp[l], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.5%

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

   1. 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 71.5%

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

5. Taylor expanded in K around 0

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

   1. cos-neg N/A

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

   2. cos-lowering-cos.f64 94.4%

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

7. Simplified 94.4%

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

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 35.8%

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

11. Taylor expanded in M around 0

    \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{exp.f64}\left(\ell\right)\right) \]
12. Step-by-step derivation

13. Simplified 35.4%

    \[\leadsto \frac{\color{blue}{1}}{e^{\ell}} \]
14. Add Preprocessing

Alternative 15: 7.0% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.5%

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

   1. 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 71.5%

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

5. Taylor expanded in K around 0

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

   1. cos-neg N/A

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

   2. cos-lowering-cos.f64 94.4%

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

7. Simplified 94.4%

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

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 35.8%

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

11. Taylor expanded in l around 0

    \[\leadsto \color{blue}{\cos M} \]
12. Step-by-step derivation

    1. cos-lowering-cos.f64 7.3%

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

13. Simplified 7.3%

    \[\leadsto \color{blue}{\cos M} \]
14. Add Preprocessing

Alternative 16: 7.0% 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 71.5%

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

   1. 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 71.5%

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

5. Taylor expanded in K around 0

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

   1. cos-neg N/A

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

   2. cos-lowering-cos.f64 94.4%

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

7. Simplified 94.4%

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

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 35.8%

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

11. Taylor expanded in l around 0

    \[\leadsto \color{blue}{\cos M} \]
12. Step-by-step derivation

    1. cos-lowering-cos.f64 7.3%

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

13. Simplified 7.3%

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

14. Taylor expanded in M around 0

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

16. Simplified 7.3%

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

Reproduce

herbie shell --seed 2024145 
(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)))))))