Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.3% → 96.8%

Time: 19.6s

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: 76.3% 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, 1.0× speedup

Math

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

FPCore

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

C

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

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 = cos(m_1) * exp(((abs((n - m)) - l) - ((((n + m) / 2.0d0) - m_1) ** 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

3. Simplified 75.9%

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

5. Taylor expanded in K around 0 97.7%

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

   1. cos-neg 97.7%

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

7. Simplified 97.7%

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

8. Final simplification 97.7%

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

Alternative 2: 67.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(n - m\right) - \left(M \cdot M + \ell\right)} \cdot \cos \left(\left(n + m\right) \cdot \left(0.5 \cdot K\right) - M\right)\\ t_1 := \cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{if}\;n \leq 1.9 \cdot 10^{-208}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 2 \cdot 10^{-126}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 4.9 \cdot 10^{-89}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 185:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0
         (* (exp (- (- n m) (+ (* M M) l))) (cos (- (* (+ n m) (* 0.5 K)) M))))
        (t_1 (* (cos M) (exp (* -0.25 (* m m))))))
   (if (<= n 1.9e-208)
     t_1
     (if (<= n 2e-126)
       t_0
       (if (<= n 4.9e-89)
         t_1
         (if (<= n 185.0) t_0 (exp (* -0.25 (* n n)))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(((n - m) - ((M * M) + l))) * cos((((n + m) * (0.5 * K)) - M));
	double t_1 = cos(M) * exp((-0.25 * (m * m)));
	double tmp;
	if (n <= 1.9e-208) {
		tmp = t_1;
	} else if (n <= 2e-126) {
		tmp = t_0;
	} else if (n <= 4.9e-89) {
		tmp = t_1;
	} else if (n <= 185.0) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(((n - m) - ((m_1 * m_1) + l))) * cos((((n + m) * (0.5d0 * k)) - m_1))
    t_1 = cos(m_1) * exp(((-0.25d0) * (m * m)))
    if (n <= 1.9d-208) then
        tmp = t_1
    else if (n <= 2d-126) then
        tmp = t_0
    else if (n <= 4.9d-89) then
        tmp = t_1
    else if (n <= 185.0d0) then
        tmp = t_0
    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 t_0 = Math.exp(((n - m) - ((M * M) + l))) * Math.cos((((n + m) * (0.5 * K)) - M));
	double t_1 = Math.cos(M) * Math.exp((-0.25 * (m * m)));
	double tmp;
	if (n <= 1.9e-208) {
		tmp = t_1;
	} else if (n <= 2e-126) {
		tmp = t_0;
	} else if (n <= 4.9e-89) {
		tmp = t_1;
	} else if (n <= 185.0) {
		tmp = t_0;
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp(((n - m) - ((M * M) + l))) * math.cos((((n + m) * (0.5 * K)) - M))
	t_1 = math.cos(M) * math.exp((-0.25 * (m * m)))
	tmp = 0
	if n <= 1.9e-208:
		tmp = t_1
	elif n <= 2e-126:
		tmp = t_0
	elif n <= 4.9e-89:
		tmp = t_1
	elif n <= 185.0:
		tmp = t_0
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(exp(Float64(Float64(n - m) - Float64(Float64(M * M) + l))) * cos(Float64(Float64(Float64(n + m) * Float64(0.5 * K)) - M)))
	t_1 = Float64(cos(M) * exp(Float64(-0.25 * Float64(m * m))))
	tmp = 0.0
	if (n <= 1.9e-208)
		tmp = t_1;
	elseif (n <= 2e-126)
		tmp = t_0;
	elseif (n <= 4.9e-89)
		tmp = t_1;
	elseif (n <= 185.0)
		tmp = t_0;
	else
		tmp = exp(Float64(-0.25 * Float64(n * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(((n - m) - ((M * M) + l))) * cos((((n + m) * (0.5 * K)) - M));
	t_1 = cos(M) * exp((-0.25 * (m * m)));
	tmp = 0.0;
	if (n <= 1.9e-208)
		tmp = t_1;
	elseif (n <= 2e-126)
		tmp = t_0;
	elseif (n <= 4.9e-89)
		tmp = t_1;
	elseif (n <= 185.0)
		tmp = t_0;
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Exp[N[(N[(n - m), $MachinePrecision] - N[(N[(M * M), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(N[(n + m), $MachinePrecision] * N[(0.5 * K), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, 1.9e-208], t$95$1, If[LessEqual[n, 2e-126], t$95$0, If[LessEqual[n, 4.9e-89], t$95$1, If[LessEqual[n, 185.0], t$95$0, N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < 1.90000000000000006e-208 or 1.9999999999999999e-126 < n < 4.9e-89

   1. Initial program 76.0%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in K around 0 96.8%

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

      1. cos-neg 96.8%

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

   7. Simplified 96.8%

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

   8. Taylor expanded in m around inf 59.1%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
   9. Step-by-step derivation

      1. unpow2 59.1%

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

   10. Simplified 59.1%

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

   if 1.90000000000000006e-208 < n < 1.9999999999999999e-126 or 4.9e-89 < n < 185

   1. Initial program 91.2%

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

   3. Simplified 91.2%

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

      1. *-commutative 91.2%

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

   6. Applied egg-rr 90.0%

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

   7. Taylor expanded in M around inf 81.3%

      \[\leadsto e^{\left(n - m\right) - \left(\color{blue}{{M}^{2}} + \ell\right)} \cdot \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right) \]
   8. Step-by-step derivation

      1. unpow2 81.3%

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

   9. Simplified 81.3%

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

   if 185 < n

   1. Initial program 68.5%

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

   3. Simplified 68.5%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in n around inf 98.7%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   9. Step-by-step derivation

      1. unpow2 98.7%

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

   10. Simplified 98.7%

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

   11. Taylor expanded in M around 0 98.7%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.9 \cdot 10^{-208}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 2 \cdot 10^{-126}:\\ \;\;\;\;e^{\left(n - m\right) - \left(M \cdot M + \ell\right)} \cdot \cos \left(\left(n + m\right) \cdot \left(0.5 \cdot K\right) - M\right)\\ \mathbf{elif}\;n \leq 4.9 \cdot 10^{-89}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 185:\\ \;\;\;\;e^{\left(n - m\right) - \left(M \cdot M + \ell\right)} \cdot \cos \left(\left(n + m\right) \cdot \left(0.5 \cdot K\right) - M\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M - \left(n + m\right) \cdot 0.5\\ \mathbf{if}\;n \leq 2 \cdot 10^{+27}:\\ \;\;\;\;e^{\left(n - m\right) - \left(\ell + t\_0 \cdot t\_0\right)} \cdot \cos \left(0.5 \cdot \left(n \cdot K\right) - M\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- M (* (+ n m) 0.5))))
   (if (<= n 2e+27)
     (* (exp (- (- n m) (+ l (* t_0 t_0)))) (cos (- (* 0.5 (* n K)) M)))
     (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = M - ((n + m) * 0.5);
	double tmp;
	if (n <= 2e+27) {
		tmp = exp(((n - m) - (l + (t_0 * t_0)))) * cos(((0.5 * (n * K)) - 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) :: t_0
    real(8) :: tmp
    t_0 = m_1 - ((n + m) * 0.5d0)
    if (n <= 2d+27) then
        tmp = exp(((n - m) - (l + (t_0 * t_0)))) * cos(((0.5d0 * (n * k)) - 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 t_0 = M - ((n + m) * 0.5);
	double tmp;
	if (n <= 2e+27) {
		tmp = Math.exp(((n - m) - (l + (t_0 * t_0)))) * Math.cos(((0.5 * (n * K)) - M));
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = M - ((n + m) * 0.5)
	tmp = 0
	if n <= 2e+27:
		tmp = math.exp(((n - m) - (l + (t_0 * t_0)))) * math.cos(((0.5 * (n * K)) - M))
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(M - Float64(Float64(n + m) * 0.5))
	tmp = 0.0
	if (n <= 2e+27)
		tmp = Float64(exp(Float64(Float64(n - m) - Float64(l + Float64(t_0 * t_0)))) * cos(Float64(Float64(0.5 * Float64(n * K)) - M)));
	else
		tmp = exp(Float64(-0.25 * Float64(n * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = M - ((n + m) * 0.5);
	tmp = 0.0;
	if (n <= 2e+27)
		tmp = exp(((n - m) - (l + (t_0 * t_0)))) * cos(((0.5 * (n * K)) - M));
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(M - N[(N[(n + m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, 2e+27], N[(N[Exp[N[(N[(n - m), $MachinePrecision] - N[(l + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(0.5 * N[(n * K), $MachinePrecision]), $MachinePrecision] - 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 < 2e27

   1. Initial program 78.4%

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

   3. Simplified 78.4%

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

      1. *-commutative 78.4%

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

   6. Applied egg-rr 78.2%

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

   7. Taylor expanded in m around 0 88.7%

      \[\leadsto e^{\left(n - m\right) - \left(\left(\left(m + n\right) \cdot 0.5 - M\right) \cdot \left(\left(m + n\right) \cdot 0.5 - M\right) + \ell\right)} \cdot \color{blue}{\cos \left(0.5 \cdot \left(K \cdot n\right) - M\right)} \]
   8. Step-by-step derivation

      1. *-commutative 88.7%

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

   9. Simplified 88.7%

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

   if 2e27 < n

   1. Initial program 68.7%

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

   3. Simplified 68.7%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in n around inf 100.0%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   9. Step-by-step derivation

      1. unpow2 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in M around 0 100.0%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2 \cdot 10^{+27}:\\ \;\;\;\;e^{\left(n - m\right) - \left(\ell + \left(M - \left(n + m\right) \cdot 0.5\right) \cdot \left(M - \left(n + m\right) \cdot 0.5\right)\right)} \cdot \cos \left(0.5 \cdot \left(n \cdot K\right) - M\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-0.25 \cdot \left(n \cdot n\right)}\\ t_1 := \cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{if}\;M \leq -1.65 \cdot 10^{-9}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;M \leq -8.5 \cdot 10^{-209}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 2.4 \cdot 10^{-222}:\\ \;\;\;\;e^{-\ell} \cdot \left(M \cdot \left(M \cdot -0.5\right)\right)\\ \mathbf{elif}\;M \leq 5000000000000:\\ \;\;\;\;t\_0 \cdot \left(1 + \left(M \cdot M\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* -0.25 (* n n)))) (t_1 (* (cos M) (exp (* M (- M))))))
   (if (<= M -1.65e-9)
     t_1
     (if (<= M -8.5e-209)
       t_0
       (if (<= M 2.4e-222)
         (* (exp (- l)) (* M (* M -0.5)))
         (if (<= M 5000000000000.0) (* t_0 (+ 1.0 (* (* M M) -0.5))) t_1))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((-0.25 * (n * n)));
	double t_1 = cos(M) * exp((M * -M));
	double tmp;
	if (M <= -1.65e-9) {
		tmp = t_1;
	} else if (M <= -8.5e-209) {
		tmp = t_0;
	} else if (M <= 2.4e-222) {
		tmp = exp(-l) * (M * (M * -0.5));
	} else if (M <= 5000000000000.0) {
		tmp = t_0 * (1.0 + ((M * M) * -0.5));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(((-0.25d0) * (n * n)))
    t_1 = cos(m_1) * exp((m_1 * -m_1))
    if (m_1 <= (-1.65d-9)) then
        tmp = t_1
    else if (m_1 <= (-8.5d-209)) then
        tmp = t_0
    else if (m_1 <= 2.4d-222) then
        tmp = exp(-l) * (m_1 * (m_1 * (-0.5d0)))
    else if (m_1 <= 5000000000000.0d0) then
        tmp = t_0 * (1.0d0 + ((m_1 * m_1) * (-0.5d0)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp((-0.25 * (n * n)));
	double t_1 = Math.cos(M) * Math.exp((M * -M));
	double tmp;
	if (M <= -1.65e-9) {
		tmp = t_1;
	} else if (M <= -8.5e-209) {
		tmp = t_0;
	} else if (M <= 2.4e-222) {
		tmp = Math.exp(-l) * (M * (M * -0.5));
	} else if (M <= 5000000000000.0) {
		tmp = t_0 * (1.0 + ((M * M) * -0.5));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp((-0.25 * (n * n)))
	t_1 = math.cos(M) * math.exp((M * -M))
	tmp = 0
	if M <= -1.65e-9:
		tmp = t_1
	elif M <= -8.5e-209:
		tmp = t_0
	elif M <= 2.4e-222:
		tmp = math.exp(-l) * (M * (M * -0.5))
	elif M <= 5000000000000.0:
		tmp = t_0 * (1.0 + ((M * M) * -0.5))
	else:
		tmp = t_1
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-0.25 * Float64(n * n)))
	t_1 = Float64(cos(M) * exp(Float64(M * Float64(-M))))
	tmp = 0.0
	if (M <= -1.65e-9)
		tmp = t_1;
	elseif (M <= -8.5e-209)
		tmp = t_0;
	elseif (M <= 2.4e-222)
		tmp = Float64(exp(Float64(-l)) * Float64(M * Float64(M * -0.5)));
	elseif (M <= 5000000000000.0)
		tmp = Float64(t_0 * Float64(1.0 + Float64(Float64(M * M) * -0.5)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp((-0.25 * (n * n)));
	t_1 = cos(M) * exp((M * -M));
	tmp = 0.0;
	if (M <= -1.65e-9)
		tmp = t_1;
	elseif (M <= -8.5e-209)
		tmp = t_0;
	elseif (M <= 2.4e-222)
		tmp = exp(-l) * (M * (M * -0.5));
	elseif (M <= 5000000000000.0)
		tmp = t_0 * (1.0 + ((M * M) * -0.5));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -1.65e-9], t$95$1, If[LessEqual[M, -8.5e-209], t$95$0, If[LessEqual[M, 2.4e-222], N[(N[Exp[(-l)], $MachinePrecision] * N[(M * N[(M * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 5000000000000.0], N[(t$95$0 * N[(1.0 + N[(N[(M * M), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if M < -1.65000000000000009e-9 or 5e12 < M

   1. Initial program 79.5%

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

   3. Simplified 79.5%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in M around inf 95.0%

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot {M}^{2}}} \]
   9. Step-by-step derivation

      1. mul-1-neg 95.0%

         \[\leadsto \cos M \cdot e^{\color{blue}{-{M}^{2}}} \]

      2. unpow2 95.0%

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

      3. distribute-rgt-neg-in 95.0%

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

   10. Simplified 95.0%

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

   if -1.65000000000000009e-9 < M < -8.5e-209

   1. Initial program 67.1%

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

   3. Simplified 67.1%

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

   5. Taylor expanded in K around 0 95.9%

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

      1. cos-neg 95.9%

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

   7. Simplified 95.9%

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

   8. Taylor expanded in n around inf 64.8%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   9. Step-by-step derivation

      1. unpow2 64.8%

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

   10. Simplified 64.8%

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

   11. Taylor expanded in M around 0 64.8%

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

   if -8.5e-209 < M < 2.39999999999999993e-222

   1. Initial program 77.0%

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

   3. Simplified 77.0%

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

   5. Taylor expanded in K around 0 97.9%

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

      1. cos-neg 97.9%

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

   7. Simplified 97.9%

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

   8. Taylor expanded in l around inf 35.6%

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   9. Step-by-step derivation

      1. mul-1-neg 35.6%

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

   10. Simplified 35.6%

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

   11. Taylor expanded in M around 0 32.9%

       \[\leadsto \color{blue}{e^{-\ell} + -0.5 \cdot \left({M}^{2} \cdot e^{-\ell}\right)} \]
   12. Step-by-step derivation

       1. associate-*r* 32.9%

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

       2. distribute-rgt1-in 35.6%

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

       3. +-commutative 35.6%

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

       4. *-commutative 35.6%

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

       5. unpow2 35.6%

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

   13. Simplified 35.6%

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

   14. Taylor expanded in M around inf 82.2%

       \[\leadsto \color{blue}{-0.5 \cdot \left({M}^{2} \cdot e^{-\ell}\right)} \]
   15. Step-by-step derivation

       1. unpow2 82.2%

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

       2. associate-*r* 82.2%

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

       3. *-commutative 82.2%

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

       4. associate-*r* 82.2%

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

   16. Simplified 82.2%

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

   if 2.39999999999999993e-222 < M < 5e12

   1. Initial program 75.0%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in K around 0 94.2%

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

      1. cos-neg 94.2%

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

   7. Simplified 94.2%

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

   8. Taylor expanded in n around inf 58.9%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   9. Step-by-step derivation

      1. unpow2 58.9%

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

   10. Simplified 58.9%

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

   11. Taylor expanded in M around 0 58.9%

       \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {M}^{2}\right)} \cdot e^{-0.25 \cdot \left(n \cdot n\right)} \]
   12. Step-by-step derivation

       1. unpow2 58.9%

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

   13. Simplified 58.9%

       \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(M \cdot M\right)\right)} \cdot e^{-0.25 \cdot \left(n \cdot n\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.65 \cdot 10^{-9}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{elif}\;M \leq -8.5 \cdot 10^{-209}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{elif}\;M \leq 2.4 \cdot 10^{-222}:\\ \;\;\;\;e^{-\ell} \cdot \left(M \cdot \left(M \cdot -0.5\right)\right)\\ \mathbf{elif}\;M \leq 5000000000000:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)} \cdot \left(1 + \left(M \cdot M\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 10^{-207}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 8.5 \cdot 10^{-74}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 1e-207)
   (* (cos M) (exp (* -0.25 (* m m))))
   (if (<= n 8.5e-74)
     (* (cos M) (exp (- l)))
     (if (<= n 54.0) (* (cos M) (exp (* M (- M)))) (exp (* -0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1e-207) {
		tmp = cos(M) * exp((-0.25 * (m * m)));
	} else if (n <= 8.5e-74) {
		tmp = cos(M) * exp(-l);
	} else if (n <= 54.0) {
		tmp = cos(M) * exp((M * -M));
	} else {
		tmp = exp((-0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 1d-207) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m * m)))
    else if (n <= 8.5d-74) then
        tmp = cos(m_1) * exp(-l)
    else if (n <= 54.0d0) then
        tmp = cos(m_1) * exp((m_1 * -m_1))
    else
        tmp = exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1e-207) {
		tmp = Math.cos(M) * Math.exp((-0.25 * (m * m)));
	} else if (n <= 8.5e-74) {
		tmp = Math.cos(M) * Math.exp(-l);
	} else if (n <= 54.0) {
		tmp = Math.cos(M) * Math.exp((M * -M));
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 1e-207:
		tmp = math.cos(M) * math.exp((-0.25 * (m * m)))
	elif n <= 8.5e-74:
		tmp = math.cos(M) * math.exp(-l)
	elif n <= 54.0:
		tmp = math.cos(M) * math.exp((M * -M))
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 1e-207)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(m * m))));
	elseif (n <= 8.5e-74)
		tmp = Float64(cos(M) * exp(Float64(-l)));
	elseif (n <= 54.0)
		tmp = Float64(cos(M) * exp(Float64(M * Float64(-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 <= 1e-207)
		tmp = cos(M) * exp((-0.25 * (m * m)));
	elseif (n <= 8.5e-74)
		tmp = cos(M) * exp(-l);
	elseif (n <= 54.0)
		tmp = cos(M) * exp((M * -M));
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 1e-207], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 8.5e-74], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 54.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < 9.99999999999999925e-208

   1. Initial program 75.3%

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

   3. Simplified 75.3%

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

   5. Taylor expanded in K around 0 97.2%

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

      1. cos-neg 97.2%

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

   7. Simplified 97.2%

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

   8. Taylor expanded in m around inf 58.0%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
   9. Step-by-step derivation

      1. unpow2 58.0%

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

   10. Simplified 58.0%

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

   if 9.99999999999999925e-208 < n < 8.50000000000000052e-74

   1. Initial program 90.1%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in K around 0 97.3%

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

      1. cos-neg 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in l around inf 54.0%

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   9. Step-by-step derivation

      1. mul-1-neg 54.0%

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

   10. Simplified 54.0%

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

   if 8.50000000000000052e-74 < n < 54

   1. Initial program 91.1%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in K around 0 91.1%

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

      1. cos-neg 91.1%

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

   7. Simplified 91.1%

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

   8. Taylor expanded in M around inf 73.4%

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot {M}^{2}}} \]
   9. Step-by-step derivation

      1. mul-1-neg 73.4%

         \[\leadsto \cos M \cdot e^{\color{blue}{-{M}^{2}}} \]

      2. unpow2 73.4%

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

      3. distribute-rgt-neg-in 73.4%

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

   10. Simplified 73.4%

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

   if 54 < n

   1. Initial program 68.9%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in n around inf 97.3%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   9. Step-by-step derivation

      1. unpow2 97.3%

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

   10. Simplified 97.3%

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

   11. Taylor expanded in M around 0 97.3%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 10^{-207}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 8.5 \cdot 10^{-74}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{elif}\;n \leq 54:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.22 \cdot 10^{-28} \lor \neg \left(n \leq 185\right):\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= n -1.22e-28) (not (<= n 185.0)))
   (exp (* -0.25 (* n n)))
   (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -1.22e-28) || !(n <= 185.0)) {
		tmp = exp((-0.25 * (n * n)));
	} else {
		tmp = cos(M) * exp(-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 ((n <= (-1.22d-28)) .or. (.not. (n <= 185.0d0))) then
        tmp = exp(((-0.25d0) * (n * n)))
    else
        tmp = cos(m_1) * exp(-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 ((n <= -1.22e-28) || !(n <= 185.0)) {
		tmp = Math.exp((-0.25 * (n * n)));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (n <= -1.22e-28) or not (n <= 185.0):
		tmp = math.exp((-0.25 * (n * n)))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((n <= -1.22e-28) || !(n <= 185.0))
		tmp = exp(Float64(-0.25 * Float64(n * n)));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((n <= -1.22e-28) || ~((n <= 185.0)))
		tmp = exp((-0.25 * (n * n)));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[n, -1.22e-28], N[Not[LessEqual[n, 185.0]], $MachinePrecision]], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.22e-28 or 185 < n

   1. Initial program 69.4%

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

   3. Simplified 69.4%

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

   5. Taylor expanded in K around 0 98.0%

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

      1. cos-neg 98.0%

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

   7. Simplified 98.0%

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

   8. Taylor expanded in n around inf 92.7%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   9. Step-by-step derivation

      1. unpow2 92.7%

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

   10. Simplified 92.7%

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

   11. Taylor expanded in M around 0 92.7%

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

   if -1.22e-28 < n < 185

   1. Initial program 84.6%

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

   3. Simplified 84.6%

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

   5. Taylor expanded in K around 0 97.4%

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

      1. cos-neg 97.4%

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

   7. Simplified 97.4%

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

   8. Taylor expanded in l around inf 44.5%

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   9. Step-by-step derivation

      1. mul-1-neg 44.5%

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

   10. Simplified 44.5%

       \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.22 \cdot 10^{-28} \lor \neg \left(n \leq 185\right):\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.22 \cdot 10^{-28} \lor \neg \left(n \leq 185\right):\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= n -1.22e-28) (not (<= n 185.0)))
   (exp (* -0.25 (* n n)))
   (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -1.22e-28) || !(n <= 185.0)) {
		tmp = exp((-0.25 * (n * n)));
	} else {
		tmp = exp(-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 ((n <= (-1.22d-28)) .or. (.not. (n <= 185.0d0))) then
        tmp = exp(((-0.25d0) * (n * n)))
    else
        tmp = exp(-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 ((n <= -1.22e-28) || !(n <= 185.0)) {
		tmp = Math.exp((-0.25 * (n * n)));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (n <= -1.22e-28) or not (n <= 185.0):
		tmp = math.exp((-0.25 * (n * n)))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((n <= -1.22e-28) || !(n <= 185.0))
		tmp = exp(Float64(-0.25 * Float64(n * n)));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((n <= -1.22e-28) || ~((n <= 185.0)))
		tmp = exp((-0.25 * (n * n)));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[n, -1.22e-28], N[Not[LessEqual[n, 185.0]], $MachinePrecision]], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.22e-28 or 185 < n

   1. Initial program 69.4%

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

   3. Simplified 69.4%

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

   5. Taylor expanded in K around 0 98.0%

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

      1. cos-neg 98.0%

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

   7. Simplified 98.0%

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

   8. Taylor expanded in n around inf 92.7%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   9. Step-by-step derivation

      1. unpow2 92.7%

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

   10. Simplified 92.7%

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

   11. Taylor expanded in M around 0 92.7%

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

   if -1.22e-28 < n < 185

   1. Initial program 84.6%

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

   3. Simplified 84.6%

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

   5. Taylor expanded in K around 0 97.4%

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

      1. cos-neg 97.4%

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

   7. Simplified 97.4%

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

   8. Taylor expanded in l around inf 44.5%

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   9. Step-by-step derivation

      1. mul-1-neg 44.5%

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

   10. Simplified 44.5%

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

   11. Taylor expanded in M around 0 43.6%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.22 \cdot 10^{-28} \lor \neg \left(n \leq 185\right):\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 37.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.02 \cdot 10^{-32}:\\ \;\;\;\;0.5 \cdot \left(K \cdot \left(m \cdot \sin M\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 1.02e-32) (* 0.5 (* K (* m (sin M)))) (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 1.02e-32) {
		tmp = 0.5 * (K * (m * sin(M)));
	} else {
		tmp = exp(-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 <= 1.02d-32) then
        tmp = 0.5d0 * (k * (m * sin(m_1)))
    else
        tmp = exp(-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 <= 1.02e-32) {
		tmp = 0.5 * (K * (m * Math.sin(M)));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 1.02e-32:
		tmp = 0.5 * (K * (m * math.sin(M)))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 1.02e-32)
		tmp = Float64(0.5 * Float64(K * Float64(m * sin(M))));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 1.02e-32)
		tmp = 0.5 * (K * (m * sin(M)));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 1.02e-32], N[(0.5 * N[(K * N[(m * N[Sin[M], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.02000000000000002e-32

   1. Initial program 74.3%

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

   3. Simplified 74.3%

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

   5. Taylor expanded in K around 0 81.3%

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

   6. Taylor expanded in l around inf 12.2%

      \[\leadsto \left(\cos \left(-M\right) + -0.5 \cdot \left(K \cdot \left(\sin \left(-M\right) \cdot \left(m + n\right)\right)\right)\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   7. Step-by-step derivation

      1. mul-1-neg 12.0%

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

   8. Simplified 12.2%

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

   9. Taylor expanded in m around inf 20.1%

      \[\leadsto \color{blue}{-0.5 \cdot \left(K \cdot \left(m \cdot \left(e^{-\ell} \cdot \sin \left(-M\right)\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 20.1%

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

       2. *-commutative 20.1%

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

       3. associate-*r* 20.1%

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

       4. rec-exp 20.1%

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

       5. associate-*r/ 20.1%

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

       6. metadata-eval 20.1%

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

       7. distribute-rgt-neg-in 20.1%

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

       8. *-commutative 20.1%

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

       9. mul-1-neg 20.1%

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

       10. remove-double-neg 20.1%

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

   11. Simplified 20.1%

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

   12. Taylor expanded in l around 0 21.7%

       \[\leadsto \color{blue}{-0.5 \cdot \left(K \cdot \left(m \cdot \sin \left(-M\right)\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* 21.7%

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

       2. *-commutative 21.7%

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

       3. remove-double-neg 21.7%

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

       4. distribute-rgt-neg-out 21.7%

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

       5. distribute-rgt-neg-in 21.7%

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

       6. *-commutative 21.7%

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

       7. distribute-rgt-neg-out 21.7%

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

       8. distribute-rgt-neg-in 21.7%

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

       9. associate-*r* 21.7%

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

       10. distribute-lft-neg-in 21.7%

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

       11. metadata-eval 21.7%

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

       12. distribute-rgt-neg-out 21.7%

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

       13. distribute-rgt-neg-in 21.7%

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

       14. sin-neg 21.7%

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

       15. distribute-rgt-neg-in 21.7%

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

       16. remove-double-neg 21.7%

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

   14. Simplified 21.7%

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

   if 1.02000000000000002e-32 < l

   1. Initial program 79.8%

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

   3. Simplified 79.8%

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

   5. Taylor expanded in K around 0 98.9%

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

      1. cos-neg 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in l around inf 92.3%

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   9. Step-by-step derivation

      1. mul-1-neg 92.3%

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

   10. Simplified 92.3%

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

   11. Taylor expanded in M around 0 92.3%

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

Alternative 9: 35.5% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

3. Simplified 75.9%

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

5. Taylor expanded in K around 0 97.7%

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

   1. cos-neg 97.7%

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

7. Simplified 97.7%

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

8. Taylor expanded in l around inf 35.2%

   \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
9. Step-by-step derivation

   1. mul-1-neg 35.2%

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

10. Simplified 35.2%

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

11. Taylor expanded in M around 0 34.5%

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

Alternative 10: 7.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \cos M \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(K, m, n, M, l):
	return math.cos(M)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

3. Simplified 75.9%

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

5. Taylor expanded in K around 0 97.7%

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

   1. cos-neg 97.7%

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

7. Simplified 97.7%

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

8. Taylor expanded in l around inf 35.2%

   \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
9. Step-by-step derivation

   1. mul-1-neg 35.2%

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

10. Simplified 35.2%

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

11. Taylor expanded in l around 0 5.6%

    \[\leadsto \color{blue}{\cos M} \]
12. Add Preprocessing

Alternative 11: 7.2% 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%

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

3. Simplified 75.9%

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

5. Taylor expanded in K around 0 97.7%

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

   1. cos-neg 97.7%

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

7. Simplified 97.7%

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

8. Taylor expanded in l around inf 35.2%

   \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
9. Step-by-step derivation

   1. mul-1-neg 35.2%

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

10. Simplified 35.2%

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

11. Taylor expanded in l around 0 5.6%

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

12. Taylor expanded in M around 0 5.6%

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

Reproduce

herbie shell --seed 2024097 
(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)))))))