Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.2% → 96.4%

Time: 14.0s

Alternatives: 10

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: 75.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.4% accurate, 1.3× 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 77.4%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
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 77.4%

   \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
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 96.7%

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

7. Simplified 96.7%

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

Alternative 2: 94.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\cos M}{e^{M \cdot M}}\\ \mathbf{if}\;M \leq -8 \cdot 10^{+75}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 86:\\ \;\;\;\;e^{\left(\left|m - n\right| - 0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (/ (cos M) (exp (* M M)))))
   (if (<= M -8e+75)
     t_0
     (if (<= M 86.0)
       (exp (- (- (fabs (- m n)) (* 0.25 (* (+ m n) (+ m n)))) l))
       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 <= -8e+75) {
		tmp = t_0;
	} else if (M <= 86.0) {
		tmp = exp(((fabs((m - n)) - (0.25 * ((m + n) * (m + n)))) - 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 = cos(m_1) / exp((m_1 * m_1))
    if (m_1 <= (-8d+75)) then
        tmp = t_0
    else if (m_1 <= 86.0d0) then
        tmp = exp(((abs((m - n)) - (0.25d0 * ((m + n) * (m + n)))) - 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 = Math.cos(M) / Math.exp((M * M));
	double tmp;
	if (M <= -8e+75) {
		tmp = t_0;
	} else if (M <= 86.0) {
		tmp = Math.exp(((Math.abs((m - n)) - (0.25 * ((m + n) * (m + n)))) - l));
	} 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 <= -8e+75:
		tmp = t_0
	elif M <= 86.0:
		tmp = math.exp(((math.fabs((m - n)) - (0.25 * ((m + n) * (m + n)))) - l))
	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 <= -8e+75)
		tmp = t_0;
	elseif (M <= 86.0)
		tmp = exp(Float64(Float64(abs(Float64(m - n)) - Float64(0.25 * Float64(Float64(m + n) * Float64(m + n)))) - l));
	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 <= -8e+75)
		tmp = t_0;
	elseif (M <= 86.0)
		tmp = exp(((abs((m - n)) - (0.25 * ((m + n) * (m + n)))) - l));
	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, -8e+75], t$95$0, If[LessEqual[M, 86.0], N[Exp[N[(N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(0.25 * N[(N[(m + n), $MachinePrecision] * N[(m + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -7.99999999999999941e75 or 86 < M

   1. Initial program 82.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 82.6%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
   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 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 99.1%

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

   10. Simplified 99.1%

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

   if -7.99999999999999941e75 < M < 86

   1. Initial program 73.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 73.6%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
   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 94.3%

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

   7. Simplified 94.3%

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

   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. rec-exp N/A

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

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

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

      3. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(\ell + \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|m - n\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|m - n\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2}\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({\left(m + n\right)}^{2}\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(m + n\right), \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

      13. fabs-sub N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. sub-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right)\right) \]

      19. --lowering--.f64 93.7%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(n, m\right)\right)\right)\right)\right)\right) \]

   10. Simplified 93.7%

       \[\leadsto \color{blue}{e^{0 - \left(\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|n - m\right|\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -8 \cdot 10^{+75}:\\ \;\;\;\;\frac{\cos M}{e^{M \cdot M}}\\ \mathbf{elif}\;M \leq 86:\\ \;\;\;\;e^{\left(\left|m - n\right| - 0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{M \cdot M}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -54:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;m \leq 1.4 \cdot 10^{-305}:\\ \;\;\;\;\frac{\cos M}{e^{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -54.0)
   (exp (* (* m m) -0.25))
   (if (<= m 1.4e-305)
     (/ (cos M) (exp (* M M)))
     (/ (cos M) (exp (* 0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -54.0) {
		tmp = exp(((m * m) * -0.25));
	} else if (m <= 1.4e-305) {
		tmp = cos(M) / exp((M * M));
	} else {
		tmp = cos(M) / exp((0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-54.0d0)) then
        tmp = exp(((m * m) * (-0.25d0)))
    else if (m <= 1.4d-305) then
        tmp = cos(m_1) / exp((m_1 * m_1))
    else
        tmp = cos(m_1) / exp((0.25d0 * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -54.0) {
		tmp = Math.exp(((m * m) * -0.25));
	} else if (m <= 1.4e-305) {
		tmp = Math.cos(M) / Math.exp((M * M));
	} else {
		tmp = Math.cos(M) / Math.exp((0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -54.0:
		tmp = math.exp(((m * m) * -0.25))
	elif m <= 1.4e-305:
		tmp = math.cos(M) / math.exp((M * M))
	else:
		tmp = math.cos(M) / math.exp((0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -54.0)
		tmp = exp(Float64(Float64(m * m) * -0.25));
	elseif (m <= 1.4e-305)
		tmp = Float64(cos(M) / exp(Float64(M * M)));
	else
		tmp = Float64(cos(M) / exp(Float64(0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -54.0)
		tmp = exp(((m * m) * -0.25));
	elseif (m <= 1.4e-305)
		tmp = cos(M) / exp((M * M));
	else
		tmp = cos(M) / exp((0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -54.0], N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, 1.4e-305], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(M * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -54

   1. Initial program 76.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 76.6%

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

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

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

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

      3. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(\ell + \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|m - n\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|m - n\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2}\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({\left(m + n\right)}^{2}\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(m + n\right), \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

      13. fabs-sub N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. sub-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right)\right) \]

      19. --lowering--.f64 95.4%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(n, m\right)\right)\right)\right)\right)\right) \]

   10. Simplified 95.4%

       \[\leadsto \color{blue}{e^{0 - \left(\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|n - m\right|\right)\right)}} \]

   11. Taylor expanded in m around inf

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

       1. *-commutative N/A

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

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

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 96.9%

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

   13. Simplified 96.9%

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

   if -54 < m < 1.40000000000000007e-305

   1. Initial program 78.8%

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

      1. neg-sub0 N/A

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

      2. associate--l- N/A

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

      3. exp-diff N/A

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

      4. associate-*r/ N/A

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

      5. exp-0 N/A

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

      6. *-rgt-identity N/A

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

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

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

   3. Simplified 78.8%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
   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 89.8%

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

   7. Simplified 89.8%

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

   8. Taylor expanded in M around 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 57.4%

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

   10. Simplified 57.4%

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

   if 1.40000000000000007e-305 < m

   1. Initial program 77.2%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 77.2%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
   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 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in 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 57.7%

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

   10. Simplified 57.7%

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

Alternative 4: 72.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.45 \cdot 10^{+22}:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.25 \cdot \left(n \cdot n\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -2.45e+22)
   (exp (* (* m m) -0.25))
   (exp (- (fabs (- m n)) (+ l (* 0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -2.45e+22) {
		tmp = exp(((m * m) * -0.25));
	} else {
		tmp = exp((fabs((m - n)) - (l + (0.25 * (n * n)))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-2.45d+22)) then
        tmp = exp(((m * m) * (-0.25d0)))
    else
        tmp = exp((abs((m - n)) - (l + (0.25d0 * (n * n)))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -2.45e+22) {
		tmp = Math.exp(((m * m) * -0.25));
	} else {
		tmp = Math.exp((Math.abs((m - n)) - (l + (0.25 * (n * n)))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -2.45e+22:
		tmp = math.exp(((m * m) * -0.25))
	else:
		tmp = math.exp((math.fabs((m - n)) - (l + (0.25 * (n * n)))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -2.45e+22)
		tmp = exp(Float64(Float64(m * m) * -0.25));
	else
		tmp = exp(Float64(abs(Float64(m - n)) - Float64(l + Float64(0.25 * Float64(n * n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -2.45e+22)
		tmp = exp(((m * m) * -0.25));
	else
		tmp = exp((abs((m - n)) - (l + (0.25 * (n * n)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -2.45e+22], N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(l + N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -2.4499999999999999e22

   1. Initial program 77.0%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. 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 77.0%

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

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

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

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

      3. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(\ell + \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|m - n\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|m - n\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2}\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({\left(m + n\right)}^{2}\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(m + n\right), \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

      13. fabs-sub N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. sub-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right)\right) \]

      19. --lowering--.f64 95.2%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(n, m\right)\right)\right)\right)\right)\right) \]

   10. Simplified 95.2%

       \[\leadsto \color{blue}{e^{0 - \left(\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|n - m\right|\right)\right)}} \]

   11. Taylor expanded in m around inf

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

       1. *-commutative N/A

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

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

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 100.0%

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

   13. Simplified 100.0%

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

   if -2.4499999999999999e22 < m

   1. Initial program 77.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. 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 77.6%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
   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 95.7%

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

   7. Simplified 95.7%

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

   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. rec-exp N/A

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

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

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

      3. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(\ell + \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|m - n\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|m - n\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2}\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({\left(m + n\right)}^{2}\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(m + n\right), \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

      13. fabs-sub N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. sub-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right)\right) \]

      19. --lowering--.f64 83.7%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(n, m\right)\right)\right)\right)\right)\right) \]

   10. Simplified 83.7%

       \[\leadsto \color{blue}{e^{0 - \left(\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|n - m\right|\right)\right)}} \]

   11. Taylor expanded in m around 0

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

       1. sub-neg N/A

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

       2. mul-1-neg N/A

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

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

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

       4. mul-1-neg N/A

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

       5. sub-neg N/A

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

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

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

       7. fabs-sub N/A

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

       8. sub-neg N/A

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

       9. mul-1-neg N/A

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

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

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

       11. mul-1-neg N/A

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

       12. sub-neg N/A

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

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

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

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

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

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

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 65.5%

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

   13. Simplified 65.5%

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

Alternative 5: 65.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -55:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;m \leq 9.2 \cdot 10^{-306}:\\ \;\;\;\;\frac{\cos M}{e^{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -55.0)
   (exp (* (* m m) -0.25))
   (if (<= m 9.2e-306) (/ (cos M) (exp (* M M))) (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -55.0) {
		tmp = exp(((m * m) * -0.25));
	} else if (m <= 9.2e-306) {
		tmp = cos(M) / exp((M * M));
	} else {
		tmp = exp((-0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-55.0d0)) then
        tmp = exp(((m * m) * (-0.25d0)))
    else if (m <= 9.2d-306) then
        tmp = cos(m_1) / exp((m_1 * m_1))
    else
        tmp = exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -55.0) {
		tmp = Math.exp(((m * m) * -0.25));
	} else if (m <= 9.2e-306) {
		tmp = Math.cos(M) / Math.exp((M * M));
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -55.0:
		tmp = math.exp(((m * m) * -0.25))
	elif m <= 9.2e-306:
		tmp = math.cos(M) / math.exp((M * M))
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -55.0)
		tmp = exp(Float64(Float64(m * m) * -0.25));
	elseif (m <= 9.2e-306)
		tmp = Float64(cos(M) / exp(Float64(M * M)));
	else
		tmp = exp(Float64(-0.25 * Float64(n * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -55.0)
		tmp = exp(((m * m) * -0.25));
	elseif (m <= 9.2e-306)
		tmp = cos(M) / exp((M * M));
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -55.0], N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, 9.2e-306], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(M * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -55

   1. Initial program 76.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 76.6%

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

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

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

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

      3. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(\ell + \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|m - n\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|m - n\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2}\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({\left(m + n\right)}^{2}\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(m + n\right), \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

      13. fabs-sub N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. sub-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right)\right) \]

      19. --lowering--.f64 95.4%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(n, m\right)\right)\right)\right)\right)\right) \]

   10. Simplified 95.4%

       \[\leadsto \color{blue}{e^{0 - \left(\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|n - m\right|\right)\right)}} \]

   11. Taylor expanded in m around inf

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

       1. *-commutative N/A

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

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

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 96.9%

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

   13. Simplified 96.9%

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

   if -55 < m < 9.19999999999999956e-306

   1. Initial program 78.8%

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

      1. neg-sub0 N/A

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

      2. associate--l- N/A

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

      3. exp-diff N/A

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

      4. associate-*r/ N/A

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

      5. exp-0 N/A

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

      6. *-rgt-identity N/A

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

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

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

   3. Simplified 78.8%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
   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 89.8%

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

   7. Simplified 89.8%

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

   8. Taylor expanded in M around 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 57.4%

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

   10. Simplified 57.4%

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

   if 9.19999999999999956e-306 < m

   1. Initial program 77.2%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 77.2%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
   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 98.6%

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

   7. Simplified 98.6%

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

   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. rec-exp N/A

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

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

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

      3. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(\ell + \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|m - n\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|m - n\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2}\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({\left(m + n\right)}^{2}\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(m + n\right), \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

      13. fabs-sub N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. sub-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right)\right) \]

      19. --lowering--.f64 89.5%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(n, m\right)\right)\right)\right)\right)\right) \]

   10. Simplified 89.5%

       \[\leadsto \color{blue}{e^{0 - \left(\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|n - m\right|\right)\right)}} \]

   11. Taylor expanded in n around inf

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

       1. *-commutative N/A

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

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

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 57.7%

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

   13. Simplified 57.7%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -55:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;m \leq 9.2 \cdot 10^{-306}:\\ \;\;\;\;\frac{\cos M}{e^{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{if}\;m \leq -9.5 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 3.9 \cdot 10^{-16}:\\ \;\;\;\;e^{0 - \ell}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* (* m m) -0.25))))
   (if (<= m -9.5e+17) t_0 (if (<= m 3.9e-16) (exp (- 0.0 l)) t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(((m * m) * -0.25));
	double tmp;
	if (m <= -9.5e+17) {
		tmp = t_0;
	} else if (m <= 3.9e-16) {
		tmp = exp((0.0 - 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 = exp(((m * m) * (-0.25d0)))
    if (m <= (-9.5d+17)) then
        tmp = t_0
    else if (m <= 3.9d-16) then
        tmp = exp((0.0d0 - 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 = Math.exp(((m * m) * -0.25));
	double tmp;
	if (m <= -9.5e+17) {
		tmp = t_0;
	} else if (m <= 3.9e-16) {
		tmp = Math.exp((0.0 - l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp(((m * m) * -0.25))
	tmp = 0
	if m <= -9.5e+17:
		tmp = t_0
	elif m <= 3.9e-16:
		tmp = math.exp((0.0 - l))
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(Float64(m * m) * -0.25))
	tmp = 0.0
	if (m <= -9.5e+17)
		tmp = t_0;
	elseif (m <= 3.9e-16)
		tmp = exp(Float64(0.0 - l));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(((m * m) * -0.25));
	tmp = 0.0;
	if (m <= -9.5e+17)
		tmp = t_0;
	elseif (m <= 3.9e-16)
		tmp = exp((0.0 - l));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[m, -9.5e+17], t$95$0, If[LessEqual[m, 3.9e-16], N[Exp[N[(0.0 - l), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -9.5e17 or 3.89999999999999977e-16 < m

   1. Initial program 77.2%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 77.2%

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

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

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

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

      3. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(\ell + \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|m - n\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|m - n\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2}\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({\left(m + n\right)}^{2}\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(m + n\right), \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

      13. fabs-sub N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. sub-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right)\right) \]

      19. --lowering--.f64 96.4%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(n, m\right)\right)\right)\right)\right)\right) \]

   10. Simplified 96.4%

       \[\leadsto \color{blue}{e^{0 - \left(\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|n - m\right|\right)\right)}} \]

   11. Taylor expanded in m around inf

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

       1. *-commutative N/A

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

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

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 96.4%

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

   13. Simplified 96.4%

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

   if -9.5e17 < m < 3.89999999999999977e-16

   1. Initial program 77.7%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 77.7%

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

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

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

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

      3. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(\ell + \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|m - n\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|m - n\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2}\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({\left(m + n\right)}^{2}\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(m + n\right), \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

      13. fabs-sub N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. sub-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right)\right) \]

      19. --lowering--.f64 75.2%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(n, m\right)\right)\right)\right)\right)\right) \]

   10. Simplified 75.2%

       \[\leadsto \color{blue}{e^{0 - \left(\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|n - m\right|\right)\right)}} \]

   11. Taylor expanded in l around inf

       \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \ell\right)}\right) \]
   12. Step-by-step derivation

       1. neg-mul-1 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(0 - \ell\right)\right) \]

       3. --lowering--.f64 42.5%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \ell\right)\right) \]

   13. Simplified 42.5%

       \[\leadsto e^{\color{blue}{0 - \ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 65.3% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if n < 54

   1. Initial program 79.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 79.6%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
   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 96.2%

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

   7. Simplified 96.2%

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

   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. rec-exp N/A

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

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

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

      3. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(\ell + \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|m - n\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|m - n\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2}\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({\left(m + n\right)}^{2}\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(m + n\right), \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

      13. fabs-sub N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. sub-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right)\right) \]

      19. --lowering--.f64 82.7%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(n, m\right)\right)\right)\right)\right)\right) \]

   10. Simplified 82.7%

       \[\leadsto \color{blue}{e^{0 - \left(\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|n - m\right|\right)\right)}} \]

   11. Taylor expanded in m around inf

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

       1. *-commutative N/A

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

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

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 55.8%

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

   13. Simplified 55.8%

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

   if 54 < n

   1. Initial program 70.5%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 70.5%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
   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 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in 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. rec-exp N/A

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

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

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

      3. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(\ell + \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|m - n\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|m - n\right|\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2}\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({\left(m + n\right)}^{2}\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(m + n\right), \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

      13. fabs-sub N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right)\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. sub-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right)\right) \]

      19. --lowering--.f64 98.4%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(n, m\right)\right)\right)\right)\right)\right) \]

   10. Simplified 98.4%

       \[\leadsto \color{blue}{e^{0 - \left(\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|n - m\right|\right)\right)}} \]

   11. Taylor expanded in n around inf

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

       1. *-commutative N/A

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

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

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 96.8%

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

   13. Simplified 96.8%

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

4. Final simplification 65.6%

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

Alternative 8: 35.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ e^{0 - \ell} \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (exp (- 0.0 l)))

C

double code(double K, double m, double n, double M, double l) {
	return exp((0.0 - l));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = exp((0.0d0 - l))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.exp((0.0 - l));
}

Python

def code(K, m, n, M, l):
	return math.exp((0.0 - l))

Julia

function code(K, m, n, M, l)
	return exp(Float64(0.0 - l))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = exp((0.0 - l));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[Exp[N[(0.0 - l), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 77.4%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
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 77.4%

   \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{\left(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell\right) - \left|m - n\right|}}} \]
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 96.7%

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

7. Simplified 96.7%

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

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. rec-exp N/A

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

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

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

   3. neg-sub0 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(0 - \left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

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

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(\left(\ell + \frac{1}{4} \cdot {\left(m + n\right)}^{2}\right) - \left|m - n\right|\right)\right)\right) \]

   5. associate--l+ N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(\ell + \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|m - n\right|\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \left(\frac{1}{4} \cdot {\left(m + n\right)}^{2} - \left|m - n\right|\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\left(\frac{1}{4} \cdot {\left(m + n\right)}^{2}\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({\left(m + n\right)}^{2}\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(m + n\right) \cdot \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(m + n\right), \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \left(m + n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|m - n\right|\right)\right)\right)\right)\right) \]

   13. fabs-sub N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n - m\right|\right)\right)\right)\right)\right) \]

   14. sub-neg N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + \left(\mathsf{neg}\left(m\right)\right)\right|\right)\right)\right)\right)\right) \]

   15. mul-1-neg N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \left(\left|n + -1 \cdot m\right|\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + -1 \cdot m\right)\right)\right)\right)\right)\right) \]

   17. mul-1-neg N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n + \left(\mathsf{neg}\left(m\right)\right)\right)\right)\right)\right)\right)\right) \]

   18. sub-neg N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\left(n - m\right)\right)\right)\right)\right)\right) \]

   19. --lowering--.f64 86.5%

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(\ell, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(n, m\right)\right)\right)\right)\right)\right) \]

10. Simplified 86.5%

    \[\leadsto \color{blue}{e^{0 - \left(\ell + \left(0.25 \cdot \left(\left(m + n\right) \cdot \left(m + n\right)\right) - \left|n - m\right|\right)\right)}} \]

11. Taylor expanded in l around inf

    \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \ell\right)}\right) \]
12. Step-by-step derivation

    1. neg-mul-1 N/A

       \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right) \]

    2. neg-sub0 N/A

       \[\leadsto \mathsf{exp.f64}\left(\left(0 - \ell\right)\right) \]

    3. --lowering--.f64 36.5%

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \ell\right)\right) \]

13. Simplified 36.5%

    \[\leadsto e^{\color{blue}{0 - \ell}} \]
14. Add Preprocessing

Alternative 9: 9.4% accurate, 47.2× speedup

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return 1.0 + (l * ((l * 0.5) + -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 + (l * ((l * 0.5d0) + (-1.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

3. Taylor expanded in l around inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), 2\right), M\right)\right), \mathsf{exp.f64}\left(\left(0 - \ell\right)\right)\right) \]

   3. --lowering--.f64 33.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), 2\right), M\right)\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \ell\right)\right)\right) \]

5. Simplified 33.5%

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

6. Taylor expanded in l around 0

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

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

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

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

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

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

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

   6. *-commutative N/A

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

   7. *-lowering-*.f64 10.7%

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

8. Simplified 10.7%

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

9. Taylor expanded in K around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \frac{1}{2}\right), -1\right)\right)\right)\right) \]
10. Step-by-step derivation

    1. cos-neg N/A

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

    2. cos-lowering-cos.f64 9.8%

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

11. Simplified 9.8%

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

12. Taylor expanded in M around 0

    \[\leadsto \color{blue}{1 + \ell \cdot \left(\frac{1}{2} \cdot \ell - 1\right)} \]
13. Step-by-step derivation

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

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

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

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

    3. sub-neg N/A

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

    4. metadata-eval N/A

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

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

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

    6. *-lowering-*.f64 9.8%

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

14. Simplified 9.8%

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

15. Final simplification 9.8%

    \[\leadsto 1 + \ell \cdot \left(\ell \cdot 0.5 + -1\right) \]
16. Add Preprocessing

Alternative 10: 7.1% 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 77.4%

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

3. Taylor expanded in n around inf

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

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 37.9%

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

5. Simplified 37.9%

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

6. Taylor expanded in n around 0

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

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

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

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

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

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

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

   4. *-commutative N/A

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

   5. *-lowering-*.f64 8.2%

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

8. Simplified 8.2%

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

9. Taylor expanded in m around 0

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

    1. cos-neg N/A

       \[\leadsto \cos M \]

    2. cos-lowering-cos.f64 8.4%

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

11. Simplified 8.4%

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

12. Taylor expanded in M around 0

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

14. Simplified 8.4%

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

Reproduce

herbie shell --seed 2024144 
(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)))))))