Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.6% → 96.4%

Time: 14.2s

Alternatives: 11

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.6% 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, 1.9× speedup

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return exp((fabs((n - m)) + (((M + (-0.5 * (n + m))) * ((0.5 * (n + m)) - M)) - 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((abs((n - m)) + (((m_1 + ((-0.5d0) * (n + m))) * ((0.5d0 * (n + m)) - m_1)) - l)))
end function

Java

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

Python

def code(K, m, n, M, l):
	return math.exp((math.fabs((n - m)) + (((M + (-0.5 * (n + m))) * ((0.5 * (n + m)) - M)) - l)))

Julia

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

MATLAB

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

Wolfram

code[K_, m_, n_, M_, l_] := N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(M + N[(-0.5 * N[(n + m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(n + m), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 77.5%

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

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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos \left(\left(0 - M\right) + \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\left(\mathsf{neg}\left(\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

   5. associate-+l- N/A

      \[\leadsto \mathsf{*.f64}\left(\cos \left(0 - \left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

   6. sub0-neg N/A

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

   7. cos-neg N/A

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

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

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

   9. sub-neg N/A

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

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

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

   11. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

   15. metadata-eval N/A

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

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

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

3. Simplified 77.5%

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

5. Taylor expanded in K around 0

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

   1. *-commutative N/A

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

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

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

7. Simplified 97.3%

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

8. Taylor expanded in M around 0

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

10. Simplified 97.7%

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

11. Final simplification 97.7%

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

Alternative 2: 92.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -2.7 \cdot 10^{+32}:\\ \;\;\;\;\frac{1}{e^{M \cdot M}}\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;e^{\left|n - m\right| + \left(-0.25 \cdot \left(\left(n + m\right) \cdot \left(n + m\right)\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(M \cdot M\right) \cdot \left(-1 + \frac{m}{M}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= M -2.7e+32)
   (/ 1.0 (exp (* M M)))
   (if (<= M 27.0)
     (exp (+ (fabs (- n m)) (- (* -0.25 (* (+ n m) (+ n m))) l)))
     (exp (* (* M M) (+ -1.0 (/ m M)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (M <= -2.7e+32) {
		tmp = 1.0 / exp((M * M));
	} else if (M <= 27.0) {
		tmp = exp((fabs((n - m)) + ((-0.25 * ((n + m) * (n + m))) - l)));
	} else {
		tmp = exp(((M * M) * (-1.0 + (m / M))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m_1 <= (-2.7d+32)) then
        tmp = 1.0d0 / exp((m_1 * m_1))
    else if (m_1 <= 27.0d0) then
        tmp = exp((abs((n - m)) + (((-0.25d0) * ((n + m) * (n + m))) - l)))
    else
        tmp = exp(((m_1 * m_1) * ((-1.0d0) + (m / m_1))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (M <= -2.7e+32) {
		tmp = 1.0 / Math.exp((M * M));
	} else if (M <= 27.0) {
		tmp = Math.exp((Math.abs((n - m)) + ((-0.25 * ((n + m) * (n + m))) - l)));
	} else {
		tmp = Math.exp(((M * M) * (-1.0 + (m / M))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if M <= -2.7e+32:
		tmp = 1.0 / math.exp((M * M))
	elif M <= 27.0:
		tmp = math.exp((math.fabs((n - m)) + ((-0.25 * ((n + m) * (n + m))) - l)))
	else:
		tmp = math.exp(((M * M) * (-1.0 + (m / M))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (M <= -2.7e+32)
		tmp = Float64(1.0 / exp(Float64(M * M)));
	elseif (M <= 27.0)
		tmp = exp(Float64(abs(Float64(n - m)) + Float64(Float64(-0.25 * Float64(Float64(n + m) * Float64(n + m))) - l)));
	else
		tmp = exp(Float64(Float64(M * M) * Float64(-1.0 + Float64(m / M))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (M <= -2.7e+32)
		tmp = 1.0 / exp((M * M));
	elseif (M <= 27.0)
		tmp = exp((abs((n - m)) + ((-0.25 * ((n + m) * (n + m))) - l)));
	else
		tmp = exp(((M * M) * (-1.0 + (m / M))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[M, -2.7e+32], N[(1.0 / N[Exp[N[(M * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 27.0], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] + N[(N[(-0.25 * N[(N[(n + m), $MachinePrecision] * N[(n + m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(M * M), $MachinePrecision] * N[(-1.0 + N[(m / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if M < -2.70000000000000013e32

   1. Initial program 86.0%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\left(0 - M\right) + \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\left(\mathsf{neg}\left(\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(0 - \left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      6. sub0-neg N/A

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

      7. cos-neg N/A

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

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

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

      9. sub-neg N/A

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

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

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

      11. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

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

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

   3. Simplified 86.0%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

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

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

   7. Simplified 100.0%

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

   8. Taylor expanded in M around 0

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

   10. Simplified 100.0%

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

   11. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left({M}^{2}\right)\right)\right), 1\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(0 - {M}^{2}\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left({M}^{2}\right)\right)\right), 1\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(M \cdot M\right)\right)\right), 1\right) \]

       5. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(M, M\right)\right)\right), 1\right) \]

   13. Simplified 100.0%

       \[\leadsto e^{\color{blue}{0 - M \cdot M}} \cdot 1 \]
   14. Step-by-step derivation

       1. *-rgt-identity N/A

          \[\leadsto e^{0 - M \cdot M} \]

       2. exp-diff N/A

          \[\leadsto \frac{e^{0}}{\color{blue}{e^{M \cdot M}}} \]

       3. 1-exp N/A

          \[\leadsto \frac{1}{e^{\color{blue}{M \cdot M}}} \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{M \cdot M}\right)}\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(M \cdot M\right)\right)\right) \]

       6. *-lowering-*.f64 100.0%

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

   15. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{1}{e^{M \cdot M}}} \]

   if -2.70000000000000013e32 < M < 27

   1. Initial program 74.6%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\left(0 - M\right) + \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\left(\mathsf{neg}\left(\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(0 - \left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      6. sub0-neg N/A

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

      7. cos-neg N/A

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

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

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

      9. sub-neg N/A

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

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

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

      11. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

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

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

   3. Simplified 74.6%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

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

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

   7. Simplified 95.6%

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

   8. Taylor expanded in M around 0

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

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

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

      2. associate--l+ N/A

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

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

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

      4. fabs-sub N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

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

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

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

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

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

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

      13. unpow2 N/A

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

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

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

      15. +-commutative N/A

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

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

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

      17. +-commutative N/A

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

      18. +-lowering-+.f64 95.5%

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

   10. Simplified 95.5%

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

   if 27 < M

   1. Initial program 77.1%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\left(0 - M\right) + \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\left(\mathsf{neg}\left(\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(0 - \left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      6. sub0-neg N/A

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

      7. cos-neg N/A

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

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

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

      9. sub-neg N/A

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

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

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

      11. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

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

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

   3. Simplified 77.1%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

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

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

   7. Simplified 98.6%

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

   8. Taylor expanded in M around 0

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

   10. Simplified 100.0%

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

   11. Taylor expanded in n around 0

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

       1. associate--l+ N/A

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

       2. fabs-sub N/A

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

       3. sub-neg N/A

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

       4. mul-1-neg N/A

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

       5. fabs-neg N/A

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

       6. associate--l+ N/A

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

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

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

       8. associate--l+ N/A

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

       9. fabs-neg N/A

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

       10. mul-1-neg N/A

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

       11. sub-neg N/A

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

       12. fabs-sub N/A

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

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

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

   13. Simplified 91.6%

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

   14. Taylor expanded in M around inf

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

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

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

       2. unpow2 N/A

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

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

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. +-commutative N/A

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

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

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

       8. /-lowering-/.f64 83.1%

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

   16. Simplified 83.1%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2.7 \cdot 10^{+32}:\\ \;\;\;\;\frac{1}{e^{M \cdot M}}\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;e^{\left|n - m\right| + \left(-0.25 \cdot \left(\left(n + m\right) \cdot \left(n + m\right)\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(M \cdot M\right) \cdot \left(-1 + \frac{m}{M}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 15500000000000:\\ \;\;\;\;e^{\left|n - m\right| + \left(\left(M + m \cdot -0.5\right) \cdot \left(m \cdot 0.5 - M\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 15500000000000.0)
   (exp (+ (fabs (- n m)) (- (* (+ M (* m -0.5)) (- (* m 0.5) M)) l)))
   (exp (* n (* n -0.25)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 15500000000000.0) {
		tmp = exp((fabs((n - m)) + (((M + (m * -0.5)) * ((m * 0.5) - M)) - l)));
	} else {
		tmp = exp((n * (n * -0.25)));
	}
	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 <= 15500000000000.0d0) then
        tmp = exp((abs((n - m)) + (((m_1 + (m * (-0.5d0))) * ((m * 0.5d0) - m_1)) - l)))
    else
        tmp = exp((n * (n * (-0.25d0))))
    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 <= 15500000000000.0) {
		tmp = Math.exp((Math.abs((n - m)) + (((M + (m * -0.5)) * ((m * 0.5) - M)) - l)));
	} else {
		tmp = Math.exp((n * (n * -0.25)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 15500000000000.0:
		tmp = math.exp((math.fabs((n - m)) + (((M + (m * -0.5)) * ((m * 0.5) - M)) - l)))
	else:
		tmp = math.exp((n * (n * -0.25)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 15500000000000.0)
		tmp = exp(Float64(abs(Float64(n - m)) + Float64(Float64(Float64(M + Float64(m * -0.5)) * Float64(Float64(m * 0.5) - M)) - l)));
	else
		tmp = exp(Float64(n * Float64(n * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 15500000000000.0)
		tmp = exp((abs((n - m)) + (((M + (m * -0.5)) * ((m * 0.5) - M)) - l)));
	else
		tmp = exp((n * (n * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 15500000000000.0], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(M + N[(m * -0.5), $MachinePrecision]), $MachinePrecision] * N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(n * N[(n * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 1.55e13

   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. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\left(0 - M\right) + \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\left(\mathsf{neg}\left(\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(0 - \left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      6. sub0-neg N/A

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

      7. cos-neg N/A

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

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

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

      9. sub-neg N/A

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

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

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

      11. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

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

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

   3. Simplified 77.0%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

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

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

   7. Simplified 96.6%

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

   8. Taylor expanded in M around 0

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

   10. Simplified 97.1%

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

   11. Taylor expanded in n around 0

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

       1. associate--l+ N/A

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

       2. fabs-sub N/A

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

       3. sub-neg N/A

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

       4. mul-1-neg N/A

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

       5. fabs-neg N/A

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

       6. associate--l+ N/A

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

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

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

       8. associate--l+ N/A

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

       9. fabs-neg N/A

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

       10. mul-1-neg N/A

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

       11. sub-neg N/A

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

       12. fabs-sub N/A

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

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

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

   13. Simplified 87.2%

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

   if 1.55e13 < n

   1. Initial program 79.2%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\left(0 - M\right) + \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\left(\mathsf{neg}\left(\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(0 - \left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      6. sub0-neg N/A

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

      7. cos-neg N/A

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

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

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

      9. sub-neg N/A

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

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

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

      11. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

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

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

   3. Simplified 79.2%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

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

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

   7. Simplified 100.0%

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

   8. Taylor expanded in M around 0

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

   10. Simplified 100.0%

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

   11. Taylor expanded in n around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 15500000000000:\\ \;\;\;\;e^{\left|n - m\right| + \left(\left(M + m \cdot -0.5\right) \cdot \left(m \cdot 0.5 - M\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 0.0007:\\ \;\;\;\;e^{\left|n - m\right| + \left(m \cdot \left(M + m \cdot -0.25\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)} \cdot \cos M\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 0.0007)
   (exp (+ (fabs (- n m)) (- (* m (+ M (* m -0.25))) l)))
   (* (exp (* -0.25 (* n n))) (cos M))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 0.0007) {
		tmp = exp((fabs((n - m)) + ((m * (M + (m * -0.25))) - l)));
	} else {
		tmp = exp((-0.25 * (n * n))) * cos(M);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 0.0007d0) then
        tmp = exp((abs((n - m)) + ((m * (m_1 + (m * (-0.25d0)))) - l)))
    else
        tmp = exp(((-0.25d0) * (n * n))) * cos(m_1)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 0.0007) {
		tmp = Math.exp((Math.abs((n - m)) + ((m * (M + (m * -0.25))) - l)));
	} else {
		tmp = Math.exp((-0.25 * (n * n))) * Math.cos(M);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 0.0007:
		tmp = math.exp((math.fabs((n - m)) + ((m * (M + (m * -0.25))) - l)))
	else:
		tmp = math.exp((-0.25 * (n * n))) * math.cos(M)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 0.0007)
		tmp = exp(Float64(abs(Float64(n - m)) + Float64(Float64(m * Float64(M + Float64(m * -0.25))) - l)));
	else
		tmp = Float64(exp(Float64(-0.25 * Float64(n * n))) * cos(M));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 0.0007)
		tmp = exp((abs((n - m)) + ((m * (M + (m * -0.25))) - l)));
	else
		tmp = exp((-0.25 * (n * n))) * cos(M);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 0.0007], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] + N[(N[(m * N[(M + N[(m * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 6.99999999999999993e-4

   1. Initial program 76.8%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\left(0 - M\right) + \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\left(\mathsf{neg}\left(\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(0 - \left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      6. sub0-neg N/A

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

      7. cos-neg N/A

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

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

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

      9. sub-neg N/A

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

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

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

      11. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

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

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

   3. Simplified 76.8%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

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

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

   7. Simplified 96.5%

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

   8. Taylor expanded in M around 0

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

   10. Simplified 97.0%

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

   11. Taylor expanded in n around 0

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

       1. associate--l+ N/A

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

       2. fabs-sub N/A

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

       3. sub-neg N/A

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

       4. mul-1-neg N/A

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

       5. fabs-neg N/A

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

       6. associate--l+ N/A

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

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

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

       8. associate--l+ N/A

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

       9. fabs-neg N/A

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

       10. mul-1-neg N/A

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

       11. sub-neg N/A

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

       12. fabs-sub N/A

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

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

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

   13. Simplified 87.0%

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

   14. Taylor expanded in M around 0

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

       1. associate--l+ N/A

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

       2. +-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. distribute-rgt-in N/A

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

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

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

       7. fabs-sub N/A

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

       8. sub-neg N/A

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

       9. mul-1-neg N/A

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

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

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

       11. mul-1-neg N/A

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

       12. sub-neg N/A

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

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

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

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

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

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

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

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

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

       17. *-commutative N/A

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

       18. *-lowering-*.f64 67.0%

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

   16. Simplified 67.0%

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

   if 6.99999999999999993e-4 < n

   1. Initial program 80.0%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\left(0 - M\right) + \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\left(\mathsf{neg}\left(\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(0 - \left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      6. sub0-neg N/A

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

      7. cos-neg N/A

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

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

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

      9. sub-neg N/A

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

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

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

      11. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

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

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

   3. Simplified 80.0%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

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

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

   7. Simplified 100.0%

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

   8. Taylor expanded in n around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 96.4%

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

   10. Simplified 96.4%

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

Alternative 5: 62.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 10^{-142}:\\ \;\;\;\;e^{\left|n - m\right| + m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;n \leq 13000000000000:\\ \;\;\;\;\frac{1}{e^{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 1e-142)
   (exp (+ (fabs (- n m)) (* m (* m -0.25))))
   (if (<= n 13000000000000.0) (/ 1.0 (exp (* M M))) (exp (* n (* n -0.25))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1e-142) {
		tmp = exp((fabs((n - m)) + (m * (m * -0.25))));
	} else if (n <= 13000000000000.0) {
		tmp = 1.0 / exp((M * M));
	} else {
		tmp = exp((n * (n * -0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 1d-142) then
        tmp = exp((abs((n - m)) + (m * (m * (-0.25d0)))))
    else if (n <= 13000000000000.0d0) then
        tmp = 1.0d0 / exp((m_1 * m_1))
    else
        tmp = exp((n * (n * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1e-142) {
		tmp = Math.exp((Math.abs((n - m)) + (m * (m * -0.25))));
	} else if (n <= 13000000000000.0) {
		tmp = 1.0 / Math.exp((M * M));
	} else {
		tmp = Math.exp((n * (n * -0.25)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 1e-142:
		tmp = math.exp((math.fabs((n - m)) + (m * (m * -0.25))))
	elif n <= 13000000000000.0:
		tmp = 1.0 / math.exp((M * M))
	else:
		tmp = math.exp((n * (n * -0.25)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 1e-142)
		tmp = exp(Float64(abs(Float64(n - m)) + Float64(m * Float64(m * -0.25))));
	elseif (n <= 13000000000000.0)
		tmp = Float64(1.0 / exp(Float64(M * M)));
	else
		tmp = exp(Float64(n * Float64(n * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 1e-142)
		tmp = exp((abs((n - m)) + (m * (m * -0.25))));
	elseif (n <= 13000000000000.0)
		tmp = 1.0 / exp((M * M));
	else
		tmp = exp((n * (n * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 1e-142], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] + N[(m * N[(m * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 13000000000000.0], N[(1.0 / N[Exp[N[(M * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(n * N[(n * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 1e-142

   1. Initial program 76.9%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\left(0 - M\right) + \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\left(\mathsf{neg}\left(\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(0 - \left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      6. sub0-neg N/A

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

      7. cos-neg N/A

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

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

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

      9. sub-neg N/A

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

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

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

      11. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

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

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

   3. Simplified 76.9%

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

   5. Taylor expanded in n around 0

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

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

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

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

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

   7. Simplified 73.6%

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

   8. Taylor expanded in m around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 43.9%

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

   10. Simplified 43.9%

       \[\leadsto \cos \left(M + -0.5 \cdot \left(K \cdot m\right)\right) \cdot e^{\left|n - m\right| + \color{blue}{-0.25 \cdot \left(m \cdot m\right)}} \]

   11. Taylor expanded in M around 0

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

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

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

       2. *-commutative N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

       7. *-lowering-*.f64 43.9%

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

   13. Simplified 43.9%

       \[\leadsto \color{blue}{\cos \left(m \cdot \left(-0.5 \cdot K\right)\right)} \cdot e^{\left|n - m\right| + -0.25 \cdot \left(m \cdot m\right)} \]

   14. Taylor expanded in K around 0

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

       1. exp-sum N/A

          \[\leadsto e^{\left|n - m\right|} \cdot \color{blue}{e^{\frac{-1}{4} \cdot {m}^{2}}} \]

       2. fabs-sub N/A

          \[\leadsto e^{\left|m - n\right|} \cdot e^{\color{blue}{\frac{-1}{4}} \cdot {m}^{2}} \]

       3. prod-exp N/A

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

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

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

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

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

       6. fabs-sub N/A

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

       7. sub-neg N/A

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

       8. mul-1-neg N/A

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

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

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

       10. mul-1-neg N/A

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

       11. sub-neg N/A

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

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

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

       13. unpow2 N/A

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

       14. associate-*r* N/A

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

       15. *-commutative N/A

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

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

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

       17. *-commutative N/A

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

       18. *-lowering-*.f64 55.0%

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

   16. Simplified 55.0%

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

   if 1e-142 < n < 1.3e13

   1. Initial program 78.0%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\left(0 - M\right) + \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\left(\mathsf{neg}\left(\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(0 - \left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      6. sub0-neg N/A

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

      7. cos-neg N/A

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

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

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

      9. sub-neg N/A

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

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

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

      11. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

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

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

   3. Simplified 78.0%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

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

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

   7. Simplified 91.7%

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

   8. Taylor expanded in M around 0

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

   10. Simplified 91.7%

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

   11. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left({M}^{2}\right)\right)\right), 1\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(0 - {M}^{2}\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left({M}^{2}\right)\right)\right), 1\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(M \cdot M\right)\right)\right), 1\right) \]

       5. *-lowering-*.f64 58.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(M, M\right)\right)\right), 1\right) \]

   13. Simplified 58.3%

       \[\leadsto e^{\color{blue}{0 - M \cdot M}} \cdot 1 \]
   14. Step-by-step derivation

       1. *-rgt-identity N/A

          \[\leadsto e^{0 - M \cdot M} \]

       2. exp-diff N/A

          \[\leadsto \frac{e^{0}}{\color{blue}{e^{M \cdot M}}} \]

       3. 1-exp N/A

          \[\leadsto \frac{1}{e^{\color{blue}{M \cdot M}}} \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{M \cdot M}\right)}\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(M \cdot M\right)\right)\right) \]

       6. *-lowering-*.f64 58.3%

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

   15. Applied egg-rr 58.3%

       \[\leadsto \color{blue}{\frac{1}{e^{M \cdot M}}} \]

   if 1.3e13 < n

   1. Initial program 79.2%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\left(0 - M\right) + \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\left(\mathsf{neg}\left(\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(0 - \left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      6. sub0-neg N/A

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

      7. cos-neg N/A

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

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

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

      9. sub-neg N/A

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

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

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

      11. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

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

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

   3. Simplified 79.2%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

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

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

   7. Simplified 100.0%

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

   8. Taylor expanded in M around 0

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

   10. Simplified 100.0%

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

   11. Taylor expanded in n around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 10^{-142}:\\ \;\;\;\;e^{\left|n - m\right| + m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;n \leq 13000000000000:\\ \;\;\;\;\frac{1}{e^{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 10^{-142}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 13000000000000:\\ \;\;\;\;\frac{1}{e^{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 1e-142)
   (* (cos M) (exp (* -0.25 (* m m))))
   (if (<= n 13000000000000.0) (/ 1.0 (exp (* M M))) (exp (* n (* n -0.25))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1e-142) {
		tmp = cos(M) * exp((-0.25 * (m * m)));
	} else if (n <= 13000000000000.0) {
		tmp = 1.0 / exp((M * M));
	} else {
		tmp = exp((n * (n * -0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 1d-142) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m * m)))
    else if (n <= 13000000000000.0d0) then
        tmp = 1.0d0 / exp((m_1 * m_1))
    else
        tmp = exp((n * (n * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1e-142) {
		tmp = Math.cos(M) * Math.exp((-0.25 * (m * m)));
	} else if (n <= 13000000000000.0) {
		tmp = 1.0 / Math.exp((M * M));
	} else {
		tmp = Math.exp((n * (n * -0.25)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 1e-142:
		tmp = math.cos(M) * math.exp((-0.25 * (m * m)))
	elif n <= 13000000000000.0:
		tmp = 1.0 / math.exp((M * M))
	else:
		tmp = math.exp((n * (n * -0.25)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 1e-142)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(m * m))));
	elseif (n <= 13000000000000.0)
		tmp = Float64(1.0 / exp(Float64(M * M)));
	else
		tmp = exp(Float64(n * Float64(n * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 1e-142)
		tmp = cos(M) * exp((-0.25 * (m * m)));
	elseif (n <= 13000000000000.0)
		tmp = 1.0 / exp((M * M));
	else
		tmp = exp((n * (n * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 1e-142], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 13000000000000.0], N[(1.0 / N[Exp[N[(M * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(n * N[(n * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 1e-142

   1. Initial program 76.9%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\left(0 - M\right) + \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\left(\mathsf{neg}\left(\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(0 - \left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      6. sub0-neg N/A

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

      7. cos-neg N/A

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

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

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

      9. sub-neg N/A

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

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

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

      11. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

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

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

   3. Simplified 76.9%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

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

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

   7. Simplified 97.4%

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

   8. Taylor expanded in m around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 61.9%

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

   10. Simplified 61.9%

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

   if 1e-142 < n < 1.3e13

   1. Initial program 78.0%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\left(0 - M\right) + \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\left(\mathsf{neg}\left(\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(0 - \left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      6. sub0-neg N/A

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

      7. cos-neg N/A

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

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

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

      9. sub-neg N/A

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

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

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

      11. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

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

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

   3. Simplified 78.0%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

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

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

   7. Simplified 91.7%

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

   8. Taylor expanded in M around 0

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

   10. Simplified 91.7%

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

   11. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left({M}^{2}\right)\right)\right), 1\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(0 - {M}^{2}\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left({M}^{2}\right)\right)\right), 1\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(M \cdot M\right)\right)\right), 1\right) \]

       5. *-lowering-*.f64 58.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(M, M\right)\right)\right), 1\right) \]

   13. Simplified 58.3%

       \[\leadsto e^{\color{blue}{0 - M \cdot M}} \cdot 1 \]
   14. Step-by-step derivation

       1. *-rgt-identity N/A

          \[\leadsto e^{0 - M \cdot M} \]

       2. exp-diff N/A

          \[\leadsto \frac{e^{0}}{\color{blue}{e^{M \cdot M}}} \]

       3. 1-exp N/A

          \[\leadsto \frac{1}{e^{\color{blue}{M \cdot M}}} \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{M \cdot M}\right)}\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(M \cdot M\right)\right)\right) \]

       6. *-lowering-*.f64 58.3%

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

   15. Applied egg-rr 58.3%

       \[\leadsto \color{blue}{\frac{1}{e^{M \cdot M}}} \]

   if 1.3e13 < n

   1. Initial program 79.2%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\left(0 - M\right) + \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\left(\mathsf{neg}\left(\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(0 - \left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      6. sub0-neg N/A

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

      7. cos-neg N/A

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

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

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

      9. sub-neg N/A

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

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

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

      11. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

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

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

   3. Simplified 79.2%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

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

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

   7. Simplified 100.0%

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

   8. Taylor expanded in M around 0

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

   10. Simplified 100.0%

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

   11. Taylor expanded in n around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 10^{-142}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 13000000000000:\\ \;\;\;\;\frac{1}{e^{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 1.5 \cdot 10^{-143}:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;n \leq 13000000000000:\\ \;\;\;\;\frac{1}{e^{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 1.5e-143)
   (exp (* m (* m -0.25)))
   (if (<= n 13000000000000.0) (/ 1.0 (exp (* M M))) (exp (* n (* n -0.25))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1.5e-143) {
		tmp = exp((m * (m * -0.25)));
	} else if (n <= 13000000000000.0) {
		tmp = 1.0 / exp((M * M));
	} else {
		tmp = exp((n * (n * -0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 1.5d-143) then
        tmp = exp((m * (m * (-0.25d0))))
    else if (n <= 13000000000000.0d0) then
        tmp = 1.0d0 / exp((m_1 * m_1))
    else
        tmp = exp((n * (n * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1.5e-143) {
		tmp = Math.exp((m * (m * -0.25)));
	} else if (n <= 13000000000000.0) {
		tmp = 1.0 / Math.exp((M * M));
	} else {
		tmp = Math.exp((n * (n * -0.25)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 1.5e-143:
		tmp = math.exp((m * (m * -0.25)))
	elif n <= 13000000000000.0:
		tmp = 1.0 / math.exp((M * M))
	else:
		tmp = math.exp((n * (n * -0.25)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 1.5e-143)
		tmp = exp(Float64(m * Float64(m * -0.25)));
	elseif (n <= 13000000000000.0)
		tmp = Float64(1.0 / exp(Float64(M * M)));
	else
		tmp = exp(Float64(n * Float64(n * -0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 1.5e-143)
		tmp = exp((m * (m * -0.25)));
	elseif (n <= 13000000000000.0)
		tmp = 1.0 / exp((M * M));
	else
		tmp = exp((n * (n * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 1.5e-143], N[Exp[N[(m * N[(m * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 13000000000000.0], N[(1.0 / N[Exp[N[(M * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(n * N[(n * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 1.49999999999999993e-143

   1. Initial program 76.9%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\left(0 - M\right) + \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\left(\mathsf{neg}\left(\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(0 - \left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      6. sub0-neg N/A

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

      7. cos-neg N/A

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

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

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

      9. sub-neg N/A

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

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

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

      11. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

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

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

   3. Simplified 76.9%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

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

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

   7. Simplified 97.4%

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

   8. Taylor expanded in M around 0

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

   10. Simplified 97.9%

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

   11. Taylor expanded in n around 0

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

       1. associate--l+ N/A

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

       2. fabs-sub N/A

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

       3. sub-neg N/A

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

       4. mul-1-neg N/A

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

       5. fabs-neg N/A

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

       6. associate--l+ N/A

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

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

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

       8. associate--l+ N/A

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

       9. fabs-neg N/A

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

       10. mul-1-neg N/A

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

       11. sub-neg N/A

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

       12. fabs-sub N/A

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

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

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

   13. Simplified 86.4%

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

   14. Taylor expanded in m around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 61.9%

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

   16. Simplified 61.9%

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

   if 1.49999999999999993e-143 < n < 1.3e13

   1. Initial program 78.0%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\left(0 - M\right) + \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\left(\mathsf{neg}\left(\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(0 - \left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      6. sub0-neg N/A

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

      7. cos-neg N/A

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

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

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

      9. sub-neg N/A

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

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

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

      11. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

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

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

   3. Simplified 78.0%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

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

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

   7. Simplified 91.7%

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

   8. Taylor expanded in M around 0

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

   10. Simplified 91.7%

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

   11. Taylor expanded in M around inf

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

       1. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left({M}^{2}\right)\right)\right), 1\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(0 - {M}^{2}\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left({M}^{2}\right)\right)\right), 1\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(M \cdot M\right)\right)\right), 1\right) \]

       5. *-lowering-*.f64 58.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(M, M\right)\right)\right), 1\right) \]

   13. Simplified 58.3%

       \[\leadsto e^{\color{blue}{0 - M \cdot M}} \cdot 1 \]
   14. Step-by-step derivation

       1. *-rgt-identity N/A

          \[\leadsto e^{0 - M \cdot M} \]

       2. exp-diff N/A

          \[\leadsto \frac{e^{0}}{\color{blue}{e^{M \cdot M}}} \]

       3. 1-exp N/A

          \[\leadsto \frac{1}{e^{\color{blue}{M \cdot M}}} \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{M \cdot M}\right)}\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(M \cdot M\right)\right)\right) \]

       6. *-lowering-*.f64 58.3%

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

   15. Applied egg-rr 58.3%

       \[\leadsto \color{blue}{\frac{1}{e^{M \cdot M}}} \]

   if 1.3e13 < n

   1. Initial program 79.2%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\left(0 - M\right) + \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\left(\mathsf{neg}\left(\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(0 - \left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      6. sub0-neg N/A

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

      7. cos-neg N/A

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

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

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

      9. sub-neg N/A

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

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

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

      11. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

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

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

   3. Simplified 79.2%

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

   5. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

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

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

   7. Simplified 100.0%

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

   8. Taylor expanded in M around 0

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

   10. Simplified 100.0%

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

   11. Taylor expanded in n around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.5 \cdot 10^{-143}:\\ \;\;\;\;e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;n \leq 13000000000000:\\ \;\;\;\;\frac{1}{e^{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{if}\;m \leq -54:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 2 \cdot 10^{-28}:\\ \;\;\;\;\frac{1}{e^{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* m (* m -0.25)))))
   (if (<= m -54.0) t_0 (if (<= m 2e-28) (/ 1.0 (exp (* M M))) 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 <= -54.0) {
		tmp = t_0;
	} else if (m <= 2e-28) {
		tmp = 1.0 / exp((M * M));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((m * (m * (-0.25d0))))
    if (m <= (-54.0d0)) then
        tmp = t_0
    else if (m <= 2d-28) then
        tmp = 1.0d0 / exp((m_1 * m_1))
    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 <= -54.0) {
		tmp = t_0;
	} else if (m <= 2e-28) {
		tmp = 1.0 / Math.exp((M * M));
	} 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 <= -54.0:
		tmp = t_0
	elif m <= 2e-28:
		tmp = 1.0 / math.exp((M * M))
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(m * Float64(m * -0.25)))
	tmp = 0.0
	if (m <= -54.0)
		tmp = t_0;
	elseif (m <= 2e-28)
		tmp = Float64(1.0 / exp(Float64(M * M)));
	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 <= -54.0)
		tmp = t_0;
	elseif (m <= 2e-28)
		tmp = 1.0 / exp((M * M));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(m * N[(m * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[m, -54.0], t$95$0, If[LessEqual[m, 2e-28], N[(1.0 / N[Exp[N[(M * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -54 or 1.99999999999999994e-28 < m

   1. Initial program 75.9%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\left(0 - M\right) + \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\left(\mathsf{neg}\left(\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(0 - \left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      6. sub0-neg N/A

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

      7. cos-neg N/A

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

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

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

      9. sub-neg N/A

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

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

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

      11. distribute-neg-frac2 N/A

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

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

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

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

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

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right), \left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), -2\right)\right)\right), \left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      16. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), -2\right)\right)\right), \mathsf{exp.f64}\left(\left(\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)\right)\right)\right) \]

   3. Simplified 75.9%

      \[\leadsto \color{blue}{\cos \left(M + \frac{K \cdot \left(m + n\right)}{-2}\right) \cdot e^{\left(\frac{m + n}{2} - M\right) \cdot \left(M + \frac{m + n}{-2}\right) + \left(\left|m - n\right| - \ell\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos M \cdot e^{\left(\left|m - n\right| + \left(M + \frac{-1}{2} \cdot \left(m + n\right)\right) \cdot \left(\frac{1}{2} \cdot \left(m + n\right) - M\right)\right) - \ell}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto e^{\left(\left|m - n\right| + \left(M + \frac{-1}{2} \cdot \left(m + n\right)\right) \cdot \left(\frac{1}{2} \cdot \left(m + n\right) - M\right)\right) - \ell} \cdot \color{blue}{\cos M} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{\left(\left|m - n\right| + \left(M + \frac{-1}{2} \cdot \left(m + n\right)\right) \cdot \left(\frac{1}{2} \cdot \left(m + n\right) - M\right)\right) - \ell}\right), \color{blue}{\cos M}\right) \]

   7. Simplified 99.3%

      \[\leadsto \color{blue}{e^{\left|n - m\right| + \left(\left(M + -0.5 \cdot \left(m + n\right)\right) \cdot \left(\left(m + n\right) \cdot 0.5 - M\right) - \ell\right)} \cdot \cos M} \]

   8. Taylor expanded in M around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(n, m\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(M, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \frac{1}{2}\right), M\right)\right), \ell\right)\right)\right), \color{blue}{1}\right) \]
   9. Step-by-step derivation

   10. Simplified 100.0%

       \[\leadsto e^{\left|n - m\right| + \left(\left(M + -0.5 \cdot \left(m + n\right)\right) \cdot \left(\left(m + n\right) \cdot 0.5 - M\right) - \ell\right)} \cdot \color{blue}{1} \]

   11. Taylor expanded in n around 0

       \[\leadsto \color{blue}{e^{\left(\left|n - m\right| + \left(M + \frac{-1}{2} \cdot m\right) \cdot \left(\frac{1}{2} \cdot m - M\right)\right) - \ell}} \]
   12. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto e^{\left|n - m\right| + \left(\left(M + \frac{-1}{2} \cdot m\right) \cdot \left(\frac{1}{2} \cdot m - M\right) - \ell\right)} \]

       2. fabs-sub N/A

          \[\leadsto e^{\left|m - n\right| + \left(\left(M + \frac{-1}{2} \cdot m\right) \cdot \left(\frac{1}{2} \cdot m - M\right) - \ell\right)} \]

       3. sub-neg N/A

          \[\leadsto e^{\left|m + \left(\mathsf{neg}\left(n\right)\right)\right| + \left(\left(M + \frac{-1}{2} \cdot m\right) \cdot \left(\frac{1}{2} \cdot m - M\right) - \ell\right)} \]

       4. mul-1-neg N/A

          \[\leadsto e^{\left|m + -1 \cdot n\right| + \left(\left(M + \frac{-1}{2} \cdot m\right) \cdot \left(\frac{1}{2} \cdot m - M\right) - \ell\right)} \]

       5. fabs-neg N/A

          \[\leadsto e^{\left|\mathsf{neg}\left(\left(m + -1 \cdot n\right)\right)\right| + \left(\left(M + \frac{-1}{2} \cdot m\right) \cdot \left(\frac{1}{2} \cdot m - M\right) - \ell\right)} \]

       6. associate--l+ N/A

          \[\leadsto e^{\left(\left|\mathsf{neg}\left(\left(m + -1 \cdot n\right)\right)\right| + \left(M + \frac{-1}{2} \cdot m\right) \cdot \left(\frac{1}{2} \cdot m - M\right)\right) - \ell} \]

       7. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\left(\left|\mathsf{neg}\left(\left(m + -1 \cdot n\right)\right)\right| + \left(M + \frac{-1}{2} \cdot m\right) \cdot \left(\frac{1}{2} \cdot m - M\right)\right) - \ell\right)\right) \]

       8. associate--l+ N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\left|\mathsf{neg}\left(\left(m + -1 \cdot n\right)\right)\right| + \left(\left(M + \frac{-1}{2} \cdot m\right) \cdot \left(\frac{1}{2} \cdot m - M\right) - \ell\right)\right)\right) \]

       9. fabs-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\left|m + -1 \cdot n\right| + \left(\left(M + \frac{-1}{2} \cdot m\right) \cdot \left(\frac{1}{2} \cdot m - M\right) - \ell\right)\right)\right) \]

       10. mul-1-neg N/A

           \[\leadsto \mathsf{exp.f64}\left(\left(\left|m + \left(\mathsf{neg}\left(n\right)\right)\right| + \left(\left(M + \frac{-1}{2} \cdot m\right) \cdot \left(\frac{1}{2} \cdot m - M\right) - \ell\right)\right)\right) \]

       11. sub-neg N/A

           \[\leadsto \mathsf{exp.f64}\left(\left(\left|m - n\right| + \left(\left(M + \frac{-1}{2} \cdot m\right) \cdot \left(\frac{1}{2} \cdot m - M\right) - \ell\right)\right)\right) \]

       12. fabs-sub N/A

           \[\leadsto \mathsf{exp.f64}\left(\left(\left|n - m\right| + \left(\left(M + \frac{-1}{2} \cdot m\right) \cdot \left(\frac{1}{2} \cdot m - M\right) - \ell\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(\left|n - m\right|\right), \left(\left(M + \frac{-1}{2} \cdot m\right) \cdot \left(\frac{1}{2} \cdot m - M\right) - \ell\right)\right)\right) \]

   13. Simplified 89.1%

       \[\leadsto \color{blue}{e^{\left|m - n\right| + \left(\left(M + m \cdot -0.5\right) \cdot \left(m \cdot 0.5 - M\right) - \ell\right)}} \]

   14. Taylor expanded in m around inf

       \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(\frac{-1}{4} \cdot {m}^{2}\right)}\right) \]
   15. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\frac{-1}{4} \cdot \left(m \cdot m\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\left(\frac{-1}{4} \cdot m\right) \cdot m\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(m \cdot \left(\frac{-1}{4} \cdot m\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(m, \left(\frac{-1}{4} \cdot m\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(m, \left(m \cdot \frac{-1}{4}\right)\right)\right) \]

       6. *-lowering-*.f64 93.3%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(m, \mathsf{*.f64}\left(m, \frac{-1}{4}\right)\right)\right) \]

   16. Simplified 93.3%

       \[\leadsto e^{\color{blue}{m \cdot \left(m \cdot -0.25\right)}} \]

   if -54 < m < 1.99999999999999994e-28

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} + \left(\mathsf{neg}\left(M\right)\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\left(\mathsf{neg}\left(M\right)\right) + \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\left(0 - M\right) + \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\left(\mathsf{neg}\left(\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(0 - \left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\mathsf{neg}\left(\left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      7. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(M - \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}}\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(M + \left(\mathsf{neg}\left(\frac{K \cdot \left(m + n\right)}{2}\right)\right)\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \left(\mathsf{neg}\left(\frac{K \cdot \left(m + n\right)}{2}\right)\right)\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \left(\frac{K \cdot \left(m + n\right)}{\mathsf{neg}\left(2\right)}\right)\right)\right), \left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\left(K \cdot \left(m + n\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right), \left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \left(m + n\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right), \left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right), \left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), -2\right)\right)\right), \left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      16. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), -2\right)\right)\right), \mathsf{exp.f64}\left(\left(\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)\right)\right)\right) \]

   3. Simplified 79.6%

      \[\leadsto \color{blue}{\cos \left(M + \frac{K \cdot \left(m + n\right)}{-2}\right) \cdot e^{\left(\frac{m + n}{2} - M\right) \cdot \left(M + \frac{m + n}{-2}\right) + \left(\left|m - n\right| - \ell\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos M \cdot e^{\left(\left|m - n\right| + \left(M + \frac{-1}{2} \cdot \left(m + n\right)\right) \cdot \left(\frac{1}{2} \cdot \left(m + n\right) - M\right)\right) - \ell}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto e^{\left(\left|m - n\right| + \left(M + \frac{-1}{2} \cdot \left(m + n\right)\right) \cdot \left(\frac{1}{2} \cdot \left(m + n\right) - M\right)\right) - \ell} \cdot \color{blue}{\cos M} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{\left(\left|m - n\right| + \left(M + \frac{-1}{2} \cdot \left(m + n\right)\right) \cdot \left(\frac{1}{2} \cdot \left(m + n\right) - M\right)\right) - \ell}\right), \color{blue}{\cos M}\right) \]

   7. Simplified 94.6%

      \[\leadsto \color{blue}{e^{\left|n - m\right| + \left(\left(M + -0.5 \cdot \left(m + n\right)\right) \cdot \left(\left(m + n\right) \cdot 0.5 - M\right) - \ell\right)} \cdot \cos M} \]

   8. Taylor expanded in M around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(n, m\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(M, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \frac{1}{2}\right), M\right)\right), \ell\right)\right)\right), \color{blue}{1}\right) \]
   9. Step-by-step derivation

   10. Simplified 94.6%

       \[\leadsto e^{\left|n - m\right| + \left(\left(M + -0.5 \cdot \left(m + n\right)\right) \cdot \left(\left(m + n\right) \cdot 0.5 - M\right) - \ell\right)} \cdot \color{blue}{1} \]

   11. Taylor expanded in M around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot {M}^{2}\right)}\right), 1\right) \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left({M}^{2}\right)\right)\right), 1\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(0 - {M}^{2}\right)\right), 1\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left({M}^{2}\right)\right)\right), 1\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(M \cdot M\right)\right)\right), 1\right) \]

       5. *-lowering-*.f64 61.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(M, M\right)\right)\right), 1\right) \]

   13. Simplified 61.7%

       \[\leadsto e^{\color{blue}{0 - M \cdot M}} \cdot 1 \]
   14. Step-by-step derivation

       1. *-rgt-identity N/A

          \[\leadsto e^{0 - M \cdot M} \]

       2. exp-diff N/A

          \[\leadsto \frac{e^{0}}{\color{blue}{e^{M \cdot M}}} \]

       3. 1-exp N/A

          \[\leadsto \frac{1}{e^{\color{blue}{M \cdot M}}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(e^{M \cdot M}\right)}\right) \]

       5. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(M \cdot M\right)\right)\right) \]

       6. *-lowering-*.f64 61.7%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(M, M\right)\right)\right) \]

   15. Applied egg-rr 61.7%

       \[\leadsto \color{blue}{\frac{1}{e^{M \cdot M}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 68.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{if}\;m \leq -4 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 1.1 \cdot 10^{-26}:\\ \;\;\;\;e^{0 - \ell}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* m (* m -0.25)))))
   (if (<= m -4e-7) t_0 (if (<= m 1.1e-26) (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 <= -4e-7) {
		tmp = t_0;
	} else if (m <= 1.1e-26) {
		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 <= (-4d-7)) then
        tmp = t_0
    else if (m <= 1.1d-26) 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 <= -4e-7) {
		tmp = t_0;
	} else if (m <= 1.1e-26) {
		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 <= -4e-7:
		tmp = t_0
	elif m <= 1.1e-26:
		tmp = math.exp((0.0 - l))
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(m * Float64(m * -0.25)))
	tmp = 0.0
	if (m <= -4e-7)
		tmp = t_0;
	elseif (m <= 1.1e-26)
		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 <= -4e-7)
		tmp = t_0;
	elseif (m <= 1.1e-26)
		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[(m * N[(m * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[m, -4e-7], t$95$0, If[LessEqual[m, 1.1e-26], N[Exp[N[(0.0 - l), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -3.9999999999999998e-7 or 1.1e-26 < m

   1. Initial program 75.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} + \left(\mathsf{neg}\left(M\right)\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\left(\mathsf{neg}\left(M\right)\right) + \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\left(0 - M\right) + \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\left(\mathsf{neg}\left(\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(0 - \left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\mathsf{neg}\left(\left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      7. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(M - \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}}\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(M + \left(\mathsf{neg}\left(\frac{K \cdot \left(m + n\right)}{2}\right)\right)\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \left(\mathsf{neg}\left(\frac{K \cdot \left(m + n\right)}{2}\right)\right)\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \left(\frac{K \cdot \left(m + n\right)}{\mathsf{neg}\left(2\right)}\right)\right)\right), \left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\left(K \cdot \left(m + n\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right), \left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \left(m + n\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right), \left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right), \left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), -2\right)\right)\right), \left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      16. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), -2\right)\right)\right), \mathsf{exp.f64}\left(\left(\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)\right)\right)\right) \]

   3. Simplified 75.6%

      \[\leadsto \color{blue}{\cos \left(M + \frac{K \cdot \left(m + n\right)}{-2}\right) \cdot e^{\left(\frac{m + n}{2} - M\right) \cdot \left(M + \frac{m + n}{-2}\right) + \left(\left|m - n\right| - \ell\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos M \cdot e^{\left(\left|m - n\right| + \left(M + \frac{-1}{2} \cdot \left(m + n\right)\right) \cdot \left(\frac{1}{2} \cdot \left(m + n\right) - M\right)\right) - \ell}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto e^{\left(\left|m - n\right| + \left(M + \frac{-1}{2} \cdot \left(m + n\right)\right) \cdot \left(\frac{1}{2} \cdot \left(m + n\right) - M\right)\right) - \ell} \cdot \color{blue}{\cos M} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{\left(\left|m - n\right| + \left(M + \frac{-1}{2} \cdot \left(m + n\right)\right) \cdot \left(\frac{1}{2} \cdot \left(m + n\right) - M\right)\right) - \ell}\right), \color{blue}{\cos M}\right) \]

   7. Simplified 98.8%

      \[\leadsto \color{blue}{e^{\left|n - m\right| + \left(\left(M + -0.5 \cdot \left(m + n\right)\right) \cdot \left(\left(m + n\right) \cdot 0.5 - M\right) - \ell\right)} \cdot \cos M} \]

   8. Taylor expanded in M around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(n, m\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(M, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(m, n\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(m, n\right), \frac{1}{2}\right), M\right)\right), \ell\right)\right)\right), \color{blue}{1}\right) \]
   9. Step-by-step derivation

   10. Simplified 99.5%

       \[\leadsto e^{\left|n - m\right| + \left(\left(M + -0.5 \cdot \left(m + n\right)\right) \cdot \left(\left(m + n\right) \cdot 0.5 - M\right) - \ell\right)} \cdot \color{blue}{1} \]

   11. Taylor expanded in n around 0

       \[\leadsto \color{blue}{e^{\left(\left|n - m\right| + \left(M + \frac{-1}{2} \cdot m\right) \cdot \left(\frac{1}{2} \cdot m - M\right)\right) - \ell}} \]
   12. Step-by-step derivation

       1. associate--l+ N/A

          \[\leadsto e^{\left|n - m\right| + \left(\left(M + \frac{-1}{2} \cdot m\right) \cdot \left(\frac{1}{2} \cdot m - M\right) - \ell\right)} \]

       2. fabs-sub N/A

          \[\leadsto e^{\left|m - n\right| + \left(\left(M + \frac{-1}{2} \cdot m\right) \cdot \left(\frac{1}{2} \cdot m - M\right) - \ell\right)} \]

       3. sub-neg N/A

          \[\leadsto e^{\left|m + \left(\mathsf{neg}\left(n\right)\right)\right| + \left(\left(M + \frac{-1}{2} \cdot m\right) \cdot \left(\frac{1}{2} \cdot m - M\right) - \ell\right)} \]

       4. mul-1-neg N/A

          \[\leadsto e^{\left|m + -1 \cdot n\right| + \left(\left(M + \frac{-1}{2} \cdot m\right) \cdot \left(\frac{1}{2} \cdot m - M\right) - \ell\right)} \]

       5. fabs-neg N/A

          \[\leadsto e^{\left|\mathsf{neg}\left(\left(m + -1 \cdot n\right)\right)\right| + \left(\left(M + \frac{-1}{2} \cdot m\right) \cdot \left(\frac{1}{2} \cdot m - M\right) - \ell\right)} \]

       6. associate--l+ N/A

          \[\leadsto e^{\left(\left|\mathsf{neg}\left(\left(m + -1 \cdot n\right)\right)\right| + \left(M + \frac{-1}{2} \cdot m\right) \cdot \left(\frac{1}{2} \cdot m - M\right)\right) - \ell} \]

       7. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\left(\left|\mathsf{neg}\left(\left(m + -1 \cdot n\right)\right)\right| + \left(M + \frac{-1}{2} \cdot m\right) \cdot \left(\frac{1}{2} \cdot m - M\right)\right) - \ell\right)\right) \]

       8. associate--l+ N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\left|\mathsf{neg}\left(\left(m + -1 \cdot n\right)\right)\right| + \left(\left(M + \frac{-1}{2} \cdot m\right) \cdot \left(\frac{1}{2} \cdot m - M\right) - \ell\right)\right)\right) \]

       9. fabs-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\left|m + -1 \cdot n\right| + \left(\left(M + \frac{-1}{2} \cdot m\right) \cdot \left(\frac{1}{2} \cdot m - M\right) - \ell\right)\right)\right) \]

       10. mul-1-neg N/A

           \[\leadsto \mathsf{exp.f64}\left(\left(\left|m + \left(\mathsf{neg}\left(n\right)\right)\right| + \left(\left(M + \frac{-1}{2} \cdot m\right) \cdot \left(\frac{1}{2} \cdot m - M\right) - \ell\right)\right)\right) \]

       11. sub-neg N/A

           \[\leadsto \mathsf{exp.f64}\left(\left(\left|m - n\right| + \left(\left(M + \frac{-1}{2} \cdot m\right) \cdot \left(\frac{1}{2} \cdot m - M\right) - \ell\right)\right)\right) \]

       12. fabs-sub N/A

           \[\leadsto \mathsf{exp.f64}\left(\left(\left|n - m\right| + \left(\left(M + \frac{-1}{2} \cdot m\right) \cdot \left(\frac{1}{2} \cdot m - M\right) - \ell\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(\left|n - m\right|\right), \left(\left(M + \frac{-1}{2} \cdot m\right) \cdot \left(\frac{1}{2} \cdot m - M\right) - \ell\right)\right)\right) \]

   13. Simplified 89.4%

       \[\leadsto \color{blue}{e^{\left|m - n\right| + \left(\left(M + m \cdot -0.5\right) \cdot \left(m \cdot 0.5 - M\right) - \ell\right)}} \]

   14. Taylor expanded in m around inf

       \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(\frac{-1}{4} \cdot {m}^{2}\right)}\right) \]
   15. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\frac{-1}{4} \cdot \left(m \cdot m\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\left(\frac{-1}{4} \cdot m\right) \cdot m\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(m \cdot \left(\frac{-1}{4} \cdot m\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(m, \left(\frac{-1}{4} \cdot m\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(m, \left(m \cdot \frac{-1}{4}\right)\right)\right) \]

       6. *-lowering-*.f64 92.8%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(m, \mathsf{*.f64}\left(m, \frac{-1}{4}\right)\right)\right) \]

   16. Simplified 92.8%

       \[\leadsto e^{\color{blue}{m \cdot \left(m \cdot -0.25\right)}} \]

   if -3.9999999999999998e-7 < m < 1.1e-26

   1. Initial program 80.0%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} + \left(\mathsf{neg}\left(M\right)\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\left(\mathsf{neg}\left(M\right)\right) + \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\left(0 - M\right) + \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\left(\mathsf{neg}\left(\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(0 - \left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(\mathsf{neg}\left(\left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      7. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\cos \left(M - \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}}\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}}\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(M + \left(\mathsf{neg}\left(\frac{K \cdot \left(m + n\right)}{2}\right)\right)\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \left(\mathsf{neg}\left(\frac{K \cdot \left(m + n\right)}{2}\right)\right)\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \left(\frac{K \cdot \left(m + n\right)}{\mathsf{neg}\left(2\right)}\right)\right)\right), \left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\left(K \cdot \left(m + n\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right), \left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \left(m + n\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right), \left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right), \left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), -2\right)\right)\right), \left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

      16. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), -2\right)\right)\right), \mathsf{exp.f64}\left(\left(\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)\right)\right)\right) \]

   3. Simplified 80.0%

      \[\leadsto \color{blue}{\cos \left(M + \frac{K \cdot \left(m + n\right)}{-2}\right) \cdot e^{\left(\frac{m + n}{2} - M\right) \cdot \left(M + \frac{m + n}{-2}\right) + \left(\left|m - n\right| - \ell\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\cos M \cdot e^{\left(\left|m - n\right| + \left(M + \frac{-1}{2} \cdot \left(m + n\right)\right) \cdot \left(\frac{1}{2} \cdot \left(m + n\right) - M\right)\right) - \ell}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto e^{\left(\left|m - n\right| + \left(M + \frac{-1}{2} \cdot \left(m + n\right)\right) \cdot \left(\frac{1}{2} \cdot \left(m + n\right) - M\right)\right) - \ell} \cdot \color{blue}{\cos M} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{\left(\left|m - n\right| + \left(M + \frac{-1}{2} \cdot \left(m + n\right)\right) \cdot \left(\frac{1}{2} \cdot \left(m + n\right) - M\right)\right) - \ell}\right), \color{blue}{\cos M}\right) \]

   7. Simplified 95.3%

      \[\leadsto \color{blue}{e^{\left|n - m\right| + \left(\left(M + -0.5 \cdot \left(m + n\right)\right) \cdot \left(\left(m + n\right) \cdot 0.5 - M\right) - \ell\right)} \cdot \cos M} \]

   8. Taylor expanded in l around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \ell\right)}\right), \mathsf{cos.f64}\left(M\right)\right) \]
   9. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right), \mathsf{cos.f64}\left(M\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(0 - \ell\right)\right), \mathsf{cos.f64}\left(M\right)\right) \]

      3. --lowering--.f64 39.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \ell\right)\right), \mathsf{cos.f64}\left(M\right)\right) \]

   10. Simplified 39.9%

       \[\leadsto e^{\color{blue}{0 - \ell}} \cdot \cos M \]

   11. Taylor expanded in M around 0

       \[\leadsto \color{blue}{e^{\mathsf{neg}\left(\ell\right)}} \]
   12. Step-by-step derivation

       1. exp-lowering-exp.f64 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 39.9%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \ell\right)\right) \]

   13. Simplified 39.9%

       \[\leadsto \color{blue}{e^{0 - \ell}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 35.2% 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 77.5%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
2. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} + \left(\mathsf{neg}\left(M\right)\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\cos \left(\left(\mathsf{neg}\left(M\right)\right) + \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

   4. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos \left(\left(0 - M\right) + \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\left(\mathsf{neg}\left(\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

   5. associate-+l- N/A

      \[\leadsto \mathsf{*.f64}\left(\cos \left(0 - \left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

   6. sub0-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\cos \left(\mathsf{neg}\left(\left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

   7. cos-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\cos \left(M - \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}}\right)\right) \]

   8. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}}\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(M + \left(\mathsf{neg}\left(\frac{K \cdot \left(m + n\right)}{2}\right)\right)\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \left(\mathsf{neg}\left(\frac{K \cdot \left(m + n\right)}{2}\right)\right)\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

   11. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \left(\frac{K \cdot \left(m + n\right)}{\mathsf{neg}\left(2\right)}\right)\right)\right), \left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\left(K \cdot \left(m + n\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right), \left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \left(m + n\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right), \left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right), \left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), -2\right)\right)\right), \left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

   16. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), -2\right)\right)\right), \mathsf{exp.f64}\left(\left(\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)\right)\right)\right) \]

3. Simplified 77.5%

   \[\leadsto \color{blue}{\cos \left(M + \frac{K \cdot \left(m + n\right)}{-2}\right) \cdot e^{\left(\frac{m + n}{2} - M\right) \cdot \left(M + \frac{m + n}{-2}\right) + \left(\left|m - n\right| - \ell\right)}} \]
4. Add Preprocessing

5. Taylor expanded in K around 0

   \[\leadsto \color{blue}{\cos M \cdot e^{\left(\left|m - n\right| + \left(M + \frac{-1}{2} \cdot \left(m + n\right)\right) \cdot \left(\frac{1}{2} \cdot \left(m + n\right) - M\right)\right) - \ell}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto e^{\left(\left|m - n\right| + \left(M + \frac{-1}{2} \cdot \left(m + n\right)\right) \cdot \left(\frac{1}{2} \cdot \left(m + n\right) - M\right)\right) - \ell} \cdot \color{blue}{\cos M} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(e^{\left(\left|m - n\right| + \left(M + \frac{-1}{2} \cdot \left(m + n\right)\right) \cdot \left(\frac{1}{2} \cdot \left(m + n\right) - M\right)\right) - \ell}\right), \color{blue}{\cos M}\right) \]

7. Simplified 97.3%

   \[\leadsto \color{blue}{e^{\left|n - m\right| + \left(\left(M + -0.5 \cdot \left(m + n\right)\right) \cdot \left(\left(m + n\right) \cdot 0.5 - M\right) - \ell\right)} \cdot \cos M} \]

8. Taylor expanded in l around inf

   \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot \ell\right)}\right), \mathsf{cos.f64}\left(M\right)\right) \]
9. Step-by-step derivation

   1. neg-mul-1 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\ell\right)\right)\right), \mathsf{cos.f64}\left(M\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(0 - \ell\right)\right), \mathsf{cos.f64}\left(M\right)\right) \]

   3. --lowering--.f64 34.3%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \ell\right)\right), \mathsf{cos.f64}\left(M\right)\right) \]

10. Simplified 34.3%

    \[\leadsto e^{\color{blue}{0 - \ell}} \cdot \cos M \]

11. Taylor expanded in M around 0

    \[\leadsto \color{blue}{e^{\mathsf{neg}\left(\ell\right)}} \]
12. Step-by-step derivation

    1. exp-lowering-exp.f64 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 34.7%

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \ell\right)\right) \]

13. Simplified 34.7%

    \[\leadsto \color{blue}{e^{0 - \ell}} \]
14. Add Preprocessing

Alternative 11: 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 77.5%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
2. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} + \left(\mathsf{neg}\left(M\right)\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\cos \left(\left(\mathsf{neg}\left(M\right)\right) + \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

   4. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos \left(\left(0 - M\right) + \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\left(\mathsf{neg}\left(\color{blue}{{\left(\frac{m + n}{2} - M\right)}^{2}}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

   5. associate-+l- N/A

      \[\leadsto \mathsf{*.f64}\left(\cos \left(0 - \left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

   6. sub0-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\cos \left(\mathsf{neg}\left(\left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

   7. cos-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\cos \left(M - \frac{K \cdot \left(m + n\right)}{2}\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}}\right)\right) \]

   8. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(M - \frac{K \cdot \left(m + n\right)}{2}\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}}\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(M + \left(\mathsf{neg}\left(\frac{K \cdot \left(m + n\right)}{2}\right)\right)\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \left(\mathsf{neg}\left(\frac{K \cdot \left(m + n\right)}{2}\right)\right)\right)\right), \left(e^{\color{blue}{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right)} - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

   11. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \left(\frac{K \cdot \left(m + n\right)}{\mathsf{neg}\left(2\right)}\right)\right)\right), \left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\left(K \cdot \left(m + n\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right), \left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \left(m + n\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right), \left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right), \left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), -2\right)\right)\right), \left(e^{\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)}\right)\right) \]

   16. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), -2\right)\right)\right), \mathsf{exp.f64}\left(\left(\left(\mathsf{neg}\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) - \left(\ell - \left|m - n\right|\right)\right)\right)\right) \]

3. Simplified 77.5%

   \[\leadsto \color{blue}{\cos \left(M + \frac{K \cdot \left(m + n\right)}{-2}\right) \cdot e^{\left(\frac{m + n}{2} - M\right) \cdot \left(M + \frac{m + n}{-2}\right) + \left(\left|m - n\right| - \ell\right)}} \]
4. Add Preprocessing

5. Taylor expanded in M around inf

   \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), -2\right)\right)\right), \mathsf{exp.f64}\left(\color{blue}{\left({M}^{2} \cdot \left(\left(\frac{m}{M} + \frac{n}{M}\right) - 1\right)\right)}\right)\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), -2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), \left(\left(\frac{m}{M} + \frac{n}{M}\right) - 1\right)\right)\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), -2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), \left(\left(\frac{m}{M} + \frac{n}{M}\right) - 1\right)\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), -2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left(\left(\frac{m}{M} + \frac{n}{M}\right) - 1\right)\right)\right)\right) \]

   4. associate--l+ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), -2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left(\frac{m}{M} + \left(\frac{n}{M} - 1\right)\right)\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), -2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\left(\frac{m}{M}\right), \left(\frac{n}{M} - 1\right)\right)\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), -2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(m, M\right), \left(\frac{n}{M} - 1\right)\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), -2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(m, M\right), \left(\frac{n}{M} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), -2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(m, M\right), \left(\frac{n}{M} + -1\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), -2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(m, M\right), \mathsf{+.f64}\left(\left(\frac{n}{M}\right), -1\right)\right)\right)\right)\right) \]

   10. /-lowering-/.f64 40.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(M, \mathsf{/.f64}\left(\mathsf{*.f64}\left(K, \mathsf{+.f64}\left(m, n\right)\right), -2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(m, M\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(n, M\right), -1\right)\right)\right)\right)\right) \]

7. Simplified 40.3%

   \[\leadsto \cos \left(M + \frac{K \cdot \left(m + n\right)}{-2}\right) \cdot e^{\color{blue}{\left(M \cdot M\right) \cdot \left(\frac{m}{M} + \left(\frac{n}{M} + -1\right)\right)}} \]

8. Taylor expanded in M around 0

   \[\leadsto \color{blue}{\cos \left(\frac{-1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)} \]
9. Step-by-step derivation

   1. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{cos.f64}\left(\left(\frac{-1}{2} \cdot \left(K \cdot \left(m + n\right)\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(K \cdot \left(m + n\right)\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(\left(m + n\right) \cdot K\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(m + n\right), K\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(n + m\right), K\right)\right)\right) \]

   6. +-lowering-+.f64 6.5%

      \[\leadsto \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(n, m\right), K\right)\right)\right) \]

10. Simplified 6.5%

    \[\leadsto \color{blue}{\cos \left(-0.5 \cdot \left(\left(n + m\right) \cdot K\right)\right)} \]

11. Taylor expanded in K around 0

    \[\leadsto \color{blue}{1} \]
12. Step-by-step derivation

13. Simplified 7.0%

    \[\leadsto \color{blue}{1} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024162 
(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)))))))