Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.4% → 96.8%

Time: 13.6s

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: 76.4% 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: 96.8% accurate, 1.3× speedup

Math

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

FPCore

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

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))) * cos(M);
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = exp((abs((n - m)) + (((m_1 + ((-0.5d0) * (n + m))) * ((0.5d0 * (n + m)) - m_1)) - l))) * cos(m_1)
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))) * Math.cos(M);
}

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))) * math.cos(M)

Julia

function code(K, m, n, M, l)
	return Float64(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))) * cos(M))
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))) * cos(M);
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(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] * N[Cos[M], $MachinePrecision]), $MachinePrecision]

Derivation

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 96.2%

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

8. Final simplification 96.2%

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

Alternative 2: 95.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos M \cdot e^{0 - M \cdot M}\\ \mathbf{if}\;M \leq -3600000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;e^{\left|n - m\right| + \left(-0.25 \cdot \left(\left(n + m\right) \cdot \left(n + m\right)\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (cos M) (exp (- 0.0 (* M M))))))
   (if (<= M -3600000000000.0)
     t_0
     (if (<= M 27.0)
       (exp (+ (fabs (- n m)) (- (* -0.25 (* (+ n m) (+ n m))) l)))
       t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(M) * exp((0.0 - (M * M)));
	double tmp;
	if (M <= -3600000000000.0) {
		tmp = t_0;
	} else if (M <= 27.0) {
		tmp = exp((fabs((n - m)) + ((-0.25 * ((n + m) * (n + m))) - l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(m_1) * exp((0.0d0 - (m_1 * m_1)))
    if (m_1 <= (-3600000000000.0d0)) then
        tmp = t_0
    else if (m_1 <= 27.0d0) then
        tmp = exp((abs((n - m)) + (((-0.25d0) * ((n + m) * (n + m))) - l)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.cos(M) * Math.exp((0.0 - (M * M)));
	double tmp;
	if (M <= -3600000000000.0) {
		tmp = t_0;
	} else if (M <= 27.0) {
		tmp = Math.exp((Math.abs((n - m)) + ((-0.25 * ((n + m) * (n + m))) - l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos(M) * math.exp((0.0 - (M * M)))
	tmp = 0
	if M <= -3600000000000.0:
		tmp = t_0
	elif M <= 27.0:
		tmp = math.exp((math.fabs((n - m)) + ((-0.25 * ((n + m) * (n + m))) - l)))
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(cos(M) * exp(Float64(0.0 - Float64(M * M))))
	tmp = 0.0
	if (M <= -3600000000000.0)
		tmp = t_0;
	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 = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos(M) * exp((0.0 - (M * M)));
	tmp = 0.0;
	if (M <= -3600000000000.0)
		tmp = t_0;
	elseif (M <= 27.0)
		tmp = exp((abs((n - m)) + ((-0.25 * ((n + m) * (n + m))) - l)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(0.0 - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -3600000000000.0], t$95$0, 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], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -3.6e12 or 27 < M

   1. Initial program 80.0%

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

      1. *-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 M around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 97.6%

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

   10. Simplified 97.6%

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

   if -3.6e12 < M < 27

   1. Initial program 72.1%

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

      1. *-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 72.1%

      \[\leadsto \color{blue}{\cos \left(M + \frac{K \cdot \left(m + n\right)}{-2}\right) \cdot e^{\left(\frac{m + n}{2} - M\right) \cdot \left(M + \frac{m + n}{-2}\right) + \left(\left|m - n\right| - \ell\right)}} \]
   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 92.7%

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

   8. Taylor expanded in M around 0

      \[\leadsto \color{blue}{e^{\left(\left|n - m\right| + \frac{-1}{4} \cdot {\left(m + n\right)}^{2}\right) - \ell}} \]
   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 92.5%

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

   10. Simplified 92.5%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -3600000000000:\\ \;\;\;\;\cos M \cdot e^{0 - M \cdot M}\\ \mathbf{elif}\;M \leq 27:\\ \;\;\;\;e^{\left|n - m\right| + \left(-0.25 \cdot \left(\left(n + m\right) \cdot \left(n + m\right)\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{0 - M \cdot M}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 7.8 \cdot 10^{-28}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;\cos M \cdot e^{0 - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 7.8e-28)
   (* (cos M) (exp (* -0.25 (* m m))))
   (if (<= n 54.0) (* (cos M) (exp (- 0.0 (* M M)))) (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 7.8e-28) {
		tmp = cos(M) * exp((-0.25 * (m * m)));
	} else if (n <= 54.0) {
		tmp = cos(M) * exp((0.0 - (M * M)));
	} else {
		tmp = exp((-0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 7.8d-28) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m * m)))
    else if (n <= 54.0d0) then
        tmp = cos(m_1) * exp((0.0d0 - (m_1 * m_1)))
    else
        tmp = exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 7.8e-28) {
		tmp = Math.cos(M) * Math.exp((-0.25 * (m * m)));
	} else if (n <= 54.0) {
		tmp = Math.cos(M) * Math.exp((0.0 - (M * M)));
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 7.8e-28:
		tmp = math.cos(M) * math.exp((-0.25 * (m * m)))
	elif n <= 54.0:
		tmp = math.cos(M) * math.exp((0.0 - (M * M)))
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 7.8e-28)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(m * m))));
	elseif (n <= 54.0)
		tmp = Float64(cos(M) * exp(Float64(0.0 - Float64(M * M))));
	else
		tmp = exp(Float64(-0.25 * Float64(n * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 7.8e-28)
		tmp = cos(M) * exp((-0.25 * (m * m)));
	elseif (n <= 54.0)
		tmp = cos(M) * exp((0.0 - (M * M)));
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 7.8e-28], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 54.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(0.0 - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 7.79999999999999998e-28

   1. Initial program 76.5%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 76.5%

      \[\leadsto \color{blue}{\cos \left(M + \frac{K \cdot \left(m + n\right)}{-2}\right) \cdot e^{\left(\frac{m + n}{2} - M\right) \cdot \left(M + \frac{m + n}{-2}\right) + \left(\left|m - n\right| - \ell\right)}} \]
   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 96.1%

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

   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 58.8%

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

   10. Simplified 58.8%

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

   if 7.79999999999999998e-28 < n < 54

   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 80.0%

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

   8. Taylor expanded in M around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 80.1%

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

   10. Simplified 80.1%

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

   if 54 < n

   1. Initial program 73.7%

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

      1. *-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 73.7%

      \[\leadsto \color{blue}{\cos \left(M + \frac{K \cdot \left(m + n\right)}{-2}\right) \cdot e^{\left(\frac{m + n}{2} - M\right) \cdot \left(M + \frac{m + n}{-2}\right) + \left(\left|m - n\right| - \ell\right)}} \]
   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 98.2%

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

   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 96.5%

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

   10. Simplified 96.5%

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

   11. Taylor expanded in n around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 93.1%

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

   13. Simplified 93.1%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 7.8 \cdot 10^{-28}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;\cos M \cdot e^{0 - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 8.2 \cdot 10^{-15}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 8.2e-15)
   (* (cos M) (exp (* -0.25 (* m m))))
   (* (cos M) (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 8.2e-15) {
		tmp = cos(M) * exp((-0.25 * (m * m)));
	} else {
		tmp = cos(M) * exp((-0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 8.2d-15) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m * m)))
    else
        tmp = cos(m_1) * exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 8.2e-15) {
		tmp = Math.cos(M) * Math.exp((-0.25 * (m * m)));
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 8.2e-15:
		tmp = math.cos(M) * math.exp((-0.25 * (m * m)))
	else:
		tmp = math.cos(M) * math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 8.2e-15)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(m * m))));
	else
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 8.2e-15)
		tmp = cos(M) * exp((-0.25 * (m * m)));
	else
		tmp = cos(M) * exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 8.2e-15], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 8.20000000000000072e-15

   1. Initial program 76.6%

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

      1. *-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 76.6%

      \[\leadsto \color{blue}{\cos \left(M + \frac{K \cdot \left(m + n\right)}{-2}\right) \cdot e^{\left(\frac{m + n}{2} - M\right) \cdot \left(M + \frac{m + n}{-2}\right) + \left(\left|m - n\right| - \ell\right)}} \]
   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 96.1%

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

   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 59.1%

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

   10. Simplified 59.1%

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

   if 8.20000000000000072e-15 < n

   1. Initial program 73.8%

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

      1. *-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 73.8%

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

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

   8. Taylor expanded in 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 87.2%

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

   10. Simplified 87.2%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 8.2 \cdot 10^{-15}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 0.48:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 0.48) (* (cos M) (exp (* -0.25 (* m m)))) (exp (* -0.25 (* n n)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 0.48) {
		tmp = cos(M) * exp((-0.25 * (m * m)));
	} else {
		tmp = exp((-0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 0.48d0) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m * m)))
    else
        tmp = exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 0.48) {
		tmp = Math.cos(M) * Math.exp((-0.25 * (m * m)));
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 0.48:
		tmp = math.cos(M) * math.exp((-0.25 * (m * m)))
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 0.48)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(m * m))));
	else
		tmp = exp(Float64(-0.25 * Float64(n * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 0.48)
		tmp = cos(M) * exp((-0.25 * (m * m)));
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 0.48], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 0.47999999999999998

   1. Initial program 76.5%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 76.5%

      \[\leadsto \color{blue}{\cos \left(M + \frac{K \cdot \left(m + n\right)}{-2}\right) \cdot e^{\left(\frac{m + n}{2} - M\right) \cdot \left(M + \frac{m + n}{-2}\right) + \left(\left|m - n\right| - \ell\right)}} \]
   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 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 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 58.2%

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

   10. Simplified 58.2%

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

   if 0.47999999999999998 < n

   1. Initial program 74.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 74.1%

      \[\leadsto \color{blue}{\cos \left(M + \frac{K \cdot \left(m + n\right)}{-2}\right) \cdot e^{\left(\frac{m + n}{2} - M\right) \cdot \left(M + \frac{m + n}{-2}\right) + \left(\left|m - n\right| - \ell\right)}} \]
   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 98.3%

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

   8. Taylor expanded in M around 0

      \[\leadsto \color{blue}{e^{\left(\left|n - m\right| + \frac{-1}{4} \cdot {\left(m + n\right)}^{2}\right) - \ell}} \]
   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 94.9%

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

   10. Simplified 94.9%

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

   11. Taylor expanded in n around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 91.5%

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

   13. Simplified 91.5%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 0.48:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 8.2 \cdot 10^{-15}:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 8.2e-15) (exp (* -0.25 (* m m))) (exp (* -0.25 (* n n)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 8.2e-15) {
		tmp = exp((-0.25 * (m * m)));
	} else {
		tmp = exp((-0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 8.2d-15) then
        tmp = exp(((-0.25d0) * (m * m)))
    else
        tmp = exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 8.2e-15) {
		tmp = Math.exp((-0.25 * (m * m)));
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 8.2e-15:
		tmp = math.exp((-0.25 * (m * m)))
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 8.2e-15)
		tmp = exp(Float64(-0.25 * Float64(m * m)));
	else
		tmp = exp(Float64(-0.25 * Float64(n * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 8.2e-15)
		tmp = exp((-0.25 * (m * m)));
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 8.2e-15], N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 8.20000000000000072e-15

   1. Initial program 76.6%

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

      1. *-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 76.6%

      \[\leadsto \color{blue}{\cos \left(M + \frac{K \cdot \left(m + n\right)}{-2}\right) \cdot e^{\left(\frac{m + n}{2} - M\right) \cdot \left(M + \frac{m + n}{-2}\right) + \left(\left|m - n\right| - \ell\right)}} \]
   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 96.1%

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

   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 83.0%

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

   10. Simplified 83.0%

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

   11. Taylor expanded in m around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 59.0%

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

   13. Simplified 59.0%

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

   if 8.20000000000000072e-15 < n

   1. Initial program 73.8%

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

      1. *-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 73.8%

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

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

   8. Taylor expanded in M around 0

      \[\leadsto \color{blue}{e^{\left(\left|n - m\right| + \frac{-1}{4} \cdot {\left(m + n\right)}^{2}\right) - \ell}} \]
   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 90.3%

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

   10. Simplified 90.3%

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

   11. Taylor expanded in n around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 87.2%

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

   13. Simplified 87.2%

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

Alternative 7: 66.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 720:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{0 - \ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 720.0) (exp (* -0.25 (* m m))) (exp (- 0.0 l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 720.0) {
		tmp = exp((-0.25 * (m * m)));
	} else {
		tmp = exp((0.0 - l));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 720.0d0) then
        tmp = exp(((-0.25d0) * (m * m)))
    else
        tmp = exp((0.0d0 - l))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 720.0) {
		tmp = Math.exp((-0.25 * (m * m)));
	} else {
		tmp = Math.exp((0.0 - l));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 720.0:
		tmp = math.exp((-0.25 * (m * m)))
	else:
		tmp = math.exp((0.0 - l))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 720.0)
		tmp = exp(Float64(-0.25 * Float64(m * m)));
	else
		tmp = exp(Float64(0.0 - l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 720.0)
		tmp = exp((-0.25 * (m * m)));
	else
		tmp = exp((0.0 - l));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 720.0], N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(0.0 - l), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 720

   1. Initial program 76.7%

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

      1. *-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 76.7%

      \[\leadsto \color{blue}{\cos \left(M + \frac{K \cdot \left(m + n\right)}{-2}\right) \cdot e^{\left(\frac{m + n}{2} - M\right) \cdot \left(M + \frac{m + n}{-2}\right) + \left(\left|m - n\right| - \ell\right)}} \]
   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 94.7%

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

   8. Taylor expanded in M around 0

      \[\leadsto \color{blue}{e^{\left(\left|n - m\right| + \frac{-1}{4} \cdot {\left(m + n\right)}^{2}\right) - \ell}} \]
   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 78.6%

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

   10. Simplified 78.6%

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

   11. Taylor expanded in m around inf

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 53.7%

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

   13. Simplified 53.7%

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

   if 720 < l

   1. Initial program 74.0%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   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 74.0%

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

      \[\leadsto \color{blue}{e^{\left(\left|n - m\right| + \frac{-1}{4} \cdot {\left(m + n\right)}^{2}\right) - \ell}} \]
   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 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in l around inf

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

       1. neg-mul-1 N/A

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

       2. neg-sub0 N/A

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

       3. --lowering--.f64 100.0%

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

   13. Simplified 100.0%

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

Alternative 8: 34.8% 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 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 96.2%

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

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 84.7%

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

10. Simplified 84.7%

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

11. Taylor expanded in l around inf

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

    1. neg-mul-1 N/A

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

    2. neg-sub0 N/A

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

    3. --lowering--.f64 37.1%

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

13. Simplified 37.1%

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

Alternative 9: 8.5% accurate, 25.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 96.2%

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

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. mul-1-neg 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 37.9%

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

10. Simplified 37.9%

    \[\leadsto e^{\color{blue}{0 - \ell}} \cdot \cos M \]

11. Taylor expanded in l around 0

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

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

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

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

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

    3. sub-neg N/A

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

    4. metadata-eval N/A

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

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

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

    6. *-commutative N/A

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

    7. *-lowering-*.f64 9.2%

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

13. Simplified 9.2%

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

14. Taylor expanded in M around 0

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

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

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

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

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

    3. unpow2 N/A

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

    4. *-lowering-*.f64 8.5%

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

16. Simplified 8.5%

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

17. Final simplification 8.5%

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

Alternative 10: 8.8% accurate, 47.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 96.2%

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

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. mul-1-neg 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 37.9%

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

10. Simplified 37.9%

    \[\leadsto e^{\color{blue}{0 - \ell}} \cdot \cos M \]

11. Taylor expanded in l around 0

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

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

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

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

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

    3. sub-neg N/A

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

    4. metadata-eval N/A

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

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

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

    6. *-commutative N/A

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

    7. *-lowering-*.f64 9.2%

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

13. Simplified 9.2%

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

14. Taylor expanded in M around 0

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

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

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

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

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

    3. sub-neg N/A

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

    4. metadata-eval N/A

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

    5. +-commutative N/A

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

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

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

    7. *-commutative N/A

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

    8. *-lowering-*.f64 8.5%

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

16. Simplified 8.5%

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

17. Final simplification 8.5%

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

Alternative 11: 6.7% accurate, 425.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

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

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. mul-1-neg 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 37.9%

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

10. Simplified 37.9%

    \[\leadsto e^{\color{blue}{0 - \ell}} \cdot \cos M \]

11. Taylor expanded in l around 0

    \[\leadsto \color{blue}{\cos M} \]
12. Step-by-step derivation

    1. cos-lowering-cos.f64 7.5%

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

13. Simplified 7.5%

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

14. Taylor expanded in M around 0

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

16. Simplified 7.5%

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

Reproduce

herbie shell --seed 2024152 
(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)))))))