Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.1% → 96.9%

Time: 29.9s

Alternatives: 12

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: 76.1% 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: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos M \cdot e^{\left|m - n\right| - \left(\ell + {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (* (cos M) (exp (- (fabs (- m n)) (+ l (pow (- (* (+ m n) 0.5) M) 2.0))))))

C

double code(double K, double m, double n, double M, double l) {
	return cos(M) * exp((fabs((m - n)) - (l + pow((((m + n) * 0.5) - 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((m - n)) - (l + ((((m + n) * 0.5d0) - 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((m - n)) - (l + Math.pow((((m + n) * 0.5) - M), 2.0))));
}

Python

def code(K, m, n, M, l):
	return math.cos(M) * math.exp((math.fabs((m - n)) - (l + math.pow((((m + n) * 0.5) - M), 2.0))))

Julia

function code(K, m, n, M, l)
	return Float64(cos(M) * exp(Float64(abs(Float64(m - n)) - Float64(l + (Float64(Float64(Float64(m + n) * 0.5) - M) ^ 2.0)))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos(M) * exp((abs((m - n)) - (l + ((((m + n) * 0.5) - M) ^ 2.0))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(l + N[Power[N[(N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.7%

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

3. Taylor expanded in K around 0 94.9%

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

   1. cos-neg 94.9%

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

   2. sub-neg 94.9%

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

   3. sub-neg 94.9%

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

   4. associate--r+ 94.9%

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

   5. *-commutative 94.9%

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

   6. associate--r+ 94.9%

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

5. Simplified 94.9%

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

6. Final simplification 94.9%

   \[\leadsto \cos M \cdot e^{\left|m - n\right| - \left(\ell + {\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2}\right)} \]
7. Add Preprocessing

Alternative 2: 87.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2.9 \cdot 10^{+42}:\\ \;\;\;\;\cos M \cdot e^{\left|m - n\right| + \left(\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{{n}^{2} \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 2.9e+42)
   (*
    (cos M)
    (exp (+ (fabs (- m n)) (- (* (- (* m 0.5) M) (- (- M (* m 0.5)) n)) l))))
   (exp (* (pow n 2.0) -0.25))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 2.9e+42) {
		tmp = cos(M) * exp((fabs((m - n)) + ((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) - l)));
	} else {
		tmp = exp((pow(n, 2.0) * -0.25));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 2.9d+42) then
        tmp = cos(m_1) * exp((abs((m - n)) + ((((m * 0.5d0) - m_1) * ((m_1 - (m * 0.5d0)) - n)) - l)))
    else
        tmp = exp(((n ** 2.0d0) * (-0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 2.9e+42) {
		tmp = Math.cos(M) * Math.exp((Math.abs((m - n)) + ((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) - l)));
	} else {
		tmp = Math.exp((Math.pow(n, 2.0) * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 2.9e+42:
		tmp = math.cos(M) * math.exp((math.fabs((m - n)) + ((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) - l)))
	else:
		tmp = math.exp((math.pow(n, 2.0) * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 2.9e+42)
		tmp = Float64(cos(M) * exp(Float64(abs(Float64(m - n)) + Float64(Float64(Float64(Float64(m * 0.5) - M) * Float64(Float64(M - Float64(m * 0.5)) - n)) - l))));
	else
		tmp = exp(Float64((n ^ 2.0) * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 2.9e+42)
		tmp = cos(M) * exp((abs((m - n)) + ((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) - l)));
	else
		tmp = exp(((n ^ 2.0) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 2.9e+42], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[Power[n, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 2.89999999999999981e42

   1. Initial program 77.4%

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

   3. Taylor expanded in K around 0 93.6%

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

      1. cos-neg 93.6%

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

      2. sub-neg 93.6%

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

      3. sub-neg 93.6%

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

      4. associate--r+ 93.6%

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

      5. *-commutative 93.6%

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

      6. associate--r+ 93.6%

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

   5. Simplified 93.6%

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

   6. Taylor expanded in n around 0 77.1%

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

      1. *-commutative 77.1%

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

      2. *-commutative 77.1%

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

      3. unpow2 77.1%

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

      4. distribute-rgt-out 84.5%

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

      5. *-commutative 84.5%

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

      6. *-commutative 84.5%

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

   8. Simplified 84.5%

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

   if 2.89999999999999981e42 < n

   1. Initial program 68.6%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. associate--r+ 100.0%

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

      5. *-commutative 100.0%

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

      6. associate--r+ 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in m around 0 78.6%

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

      1. +-commutative 78.6%

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

      2. unpow2 78.6%

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

      3. distribute-rgt-out 86.5%

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

   8. Simplified 86.5%

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

   9. Taylor expanded in M around 0 86.5%

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

   10. Taylor expanded in n around inf 100.0%

       \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   11. Step-by-step derivation

       1. *-commutative 100.0%

          \[\leadsto 1 \cdot e^{\color{blue}{{n}^{2} \cdot -0.25}} \]

   12. Simplified 100.0%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2.9 \cdot 10^{+42}:\\ \;\;\;\;\cos M \cdot e^{\left|m - n\right| + \left(\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{{n}^{2} \cdot -0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {n}^{2} \cdot -0.25\\ \mathbf{if}\;n \leq -7.5 \cdot 10^{-126}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(0.5 \cdot \left(M - m\right) - M \cdot -0.5\right) + t\_0}\\ \mathbf{elif}\;n \leq 14600000000:\\ \;\;\;\;e^{\left|m - n\right| + \left(\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (pow n 2.0) -0.25)))
   (if (<= n -7.5e-126)
     (* (cos M) (exp (+ (* n (- (* 0.5 (- M m)) (* M -0.5))) t_0)))
     (if (<= n 14600000000.0)
       (exp (+ (fabs (- m n)) (- (* (- (* n 0.5) M) (- (- M (* n 0.5)) m)) l)))
       (exp t_0)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = pow(n, 2.0) * -0.25;
	double tmp;
	if (n <= -7.5e-126) {
		tmp = cos(M) * exp(((n * ((0.5 * (M - m)) - (M * -0.5))) + t_0));
	} else if (n <= 14600000000.0) {
		tmp = exp((fabs((m - n)) + ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l)));
	} else {
		tmp = exp(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 = (n ** 2.0d0) * (-0.25d0)
    if (n <= (-7.5d-126)) then
        tmp = cos(m_1) * exp(((n * ((0.5d0 * (m_1 - m)) - (m_1 * (-0.5d0)))) + t_0))
    else if (n <= 14600000000.0d0) then
        tmp = exp((abs((m - n)) + ((((n * 0.5d0) - m_1) * ((m_1 - (n * 0.5d0)) - m)) - l)))
    else
        tmp = exp(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.pow(n, 2.0) * -0.25;
	double tmp;
	if (n <= -7.5e-126) {
		tmp = Math.cos(M) * Math.exp(((n * ((0.5 * (M - m)) - (M * -0.5))) + t_0));
	} else if (n <= 14600000000.0) {
		tmp = Math.exp((Math.abs((m - n)) + ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l)));
	} else {
		tmp = Math.exp(t_0);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.pow(n, 2.0) * -0.25
	tmp = 0
	if n <= -7.5e-126:
		tmp = math.cos(M) * math.exp(((n * ((0.5 * (M - m)) - (M * -0.5))) + t_0))
	elif n <= 14600000000.0:
		tmp = math.exp((math.fabs((m - n)) + ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l)))
	else:
		tmp = math.exp(t_0)
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64((n ^ 2.0) * -0.25)
	tmp = 0.0
	if (n <= -7.5e-126)
		tmp = Float64(cos(M) * exp(Float64(Float64(n * Float64(Float64(0.5 * Float64(M - m)) - Float64(M * -0.5))) + t_0)));
	elseif (n <= 14600000000.0)
		tmp = exp(Float64(abs(Float64(m - n)) + Float64(Float64(Float64(Float64(n * 0.5) - M) * Float64(Float64(M - Float64(n * 0.5)) - m)) - l)));
	else
		tmp = exp(t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = (n ^ 2.0) * -0.25;
	tmp = 0.0;
	if (n <= -7.5e-126)
		tmp = cos(M) * exp(((n * ((0.5 * (M - m)) - (M * -0.5))) + t_0));
	elseif (n <= 14600000000.0)
		tmp = exp((abs((m - n)) + ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l)));
	else
		tmp = exp(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Power[n, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]}, If[LessEqual[n, -7.5e-126], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(n * N[(N[(0.5 * N[(M - m), $MachinePrecision]), $MachinePrecision] - N[(M * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 14600000000.0], N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[t$95$0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.49999999999999976e-126

   1. Initial program 72.2%

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

   3. Taylor expanded in K around 0 94.5%

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

      1. cos-neg 94.5%

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

      2. sub-neg 94.5%

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

      3. sub-neg 94.5%

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

      4. associate--r+ 94.5%

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

      5. *-commutative 94.5%

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

      6. associate--r+ 94.5%

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

   5. Simplified 94.5%

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

   6. Taylor expanded in m around 0 74.2%

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

      1. +-commutative 74.2%

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

      2. unpow2 74.2%

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

      3. distribute-rgt-out 79.4%

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

   8. Simplified 79.4%

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

   9. Taylor expanded in n around inf 58.2%

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \left(n \cdot \left(-0.5 \cdot M + 0.5 \cdot \left(m - M\right)\right)\right) + -0.25 \cdot {n}^{2}}} \]

   if -7.49999999999999976e-126 < n < 1.46e10

   1. Initial program 82.7%

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

   3. Taylor expanded in K around 0 92.1%

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

      1. cos-neg 92.1%

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

      2. sub-neg 92.1%

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

      3. sub-neg 92.1%

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

      4. associate--r+ 92.1%

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

      5. *-commutative 92.1%

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

      6. associate--r+ 92.1%

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

   5. Simplified 92.1%

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

   6. Taylor expanded in m around 0 68.7%

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

      1. +-commutative 68.7%

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

      2. unpow2 68.7%

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

      3. distribute-rgt-out 70.8%

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

   8. Simplified 70.8%

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

   9. Taylor expanded in M around 0 69.7%

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

   if 1.46e10 < n

   1. Initial program 70.0%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. associate--r+ 100.0%

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

      5. *-commutative 100.0%

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

      6. associate--r+ 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in m around 0 75.2%

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

      1. +-commutative 75.2%

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

      2. unpow2 75.2%

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

      3. distribute-rgt-out 83.6%

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

   8. Simplified 83.6%

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

   9. Taylor expanded in M around 0 83.6%

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

   10. Taylor expanded in n around inf 98.4%

       \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   11. Step-by-step derivation

       1. *-commutative 98.4%

          \[\leadsto 1 \cdot e^{\color{blue}{{n}^{2} \cdot -0.25}} \]

   12. Simplified 98.4%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7.5 \cdot 10^{-126}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(0.5 \cdot \left(M - m\right) - M \cdot -0.5\right) + {n}^{2} \cdot -0.25}\\ \mathbf{elif}\;n \leq 14600000000:\\ \;\;\;\;e^{\left|m - n\right| + \left(\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{{n}^{2} \cdot -0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {n}^{2} \cdot -0.25\\ \mathbf{if}\;n \leq -2.2 \cdot 10^{-127}:\\ \;\;\;\;e^{n \cdot \left(0.5 \cdot \left(M - m\right) - M \cdot -0.5\right) + t\_0}\\ \mathbf{elif}\;n \leq 14600000000:\\ \;\;\;\;e^{\left|m - n\right| + \left(\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (pow n 2.0) -0.25)))
   (if (<= n -2.2e-127)
     (exp (+ (* n (- (* 0.5 (- M m)) (* M -0.5))) t_0))
     (if (<= n 14600000000.0)
       (exp (+ (fabs (- m n)) (- (* (- (* n 0.5) M) (- (- M (* n 0.5)) m)) l)))
       (exp t_0)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = pow(n, 2.0) * -0.25;
	double tmp;
	if (n <= -2.2e-127) {
		tmp = exp(((n * ((0.5 * (M - m)) - (M * -0.5))) + t_0));
	} else if (n <= 14600000000.0) {
		tmp = exp((fabs((m - n)) + ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l)));
	} else {
		tmp = exp(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 = (n ** 2.0d0) * (-0.25d0)
    if (n <= (-2.2d-127)) then
        tmp = exp(((n * ((0.5d0 * (m_1 - m)) - (m_1 * (-0.5d0)))) + t_0))
    else if (n <= 14600000000.0d0) then
        tmp = exp((abs((m - n)) + ((((n * 0.5d0) - m_1) * ((m_1 - (n * 0.5d0)) - m)) - l)))
    else
        tmp = exp(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.pow(n, 2.0) * -0.25;
	double tmp;
	if (n <= -2.2e-127) {
		tmp = Math.exp(((n * ((0.5 * (M - m)) - (M * -0.5))) + t_0));
	} else if (n <= 14600000000.0) {
		tmp = Math.exp((Math.abs((m - n)) + ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l)));
	} else {
		tmp = Math.exp(t_0);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.pow(n, 2.0) * -0.25
	tmp = 0
	if n <= -2.2e-127:
		tmp = math.exp(((n * ((0.5 * (M - m)) - (M * -0.5))) + t_0))
	elif n <= 14600000000.0:
		tmp = math.exp((math.fabs((m - n)) + ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l)))
	else:
		tmp = math.exp(t_0)
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64((n ^ 2.0) * -0.25)
	tmp = 0.0
	if (n <= -2.2e-127)
		tmp = exp(Float64(Float64(n * Float64(Float64(0.5 * Float64(M - m)) - Float64(M * -0.5))) + t_0));
	elseif (n <= 14600000000.0)
		tmp = exp(Float64(abs(Float64(m - n)) + Float64(Float64(Float64(Float64(n * 0.5) - M) * Float64(Float64(M - Float64(n * 0.5)) - m)) - l)));
	else
		tmp = exp(t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = (n ^ 2.0) * -0.25;
	tmp = 0.0;
	if (n <= -2.2e-127)
		tmp = exp(((n * ((0.5 * (M - m)) - (M * -0.5))) + t_0));
	elseif (n <= 14600000000.0)
		tmp = exp((abs((m - n)) + ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l)));
	else
		tmp = exp(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Power[n, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]}, If[LessEqual[n, -2.2e-127], N[Exp[N[(N[(n * N[(N[(0.5 * N[(M - m), $MachinePrecision]), $MachinePrecision] - N[(M * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 14600000000.0], N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[t$95$0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.2000000000000001e-127

   1. Initial program 72.2%

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

   3. Taylor expanded in K around 0 94.5%

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

      1. cos-neg 94.5%

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

      2. sub-neg 94.5%

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

      3. sub-neg 94.5%

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

      4. associate--r+ 94.5%

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

      5. *-commutative 94.5%

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

      6. associate--r+ 94.5%

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

   5. Simplified 94.5%

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

   6. Taylor expanded in m around 0 74.2%

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

      1. +-commutative 74.2%

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

      2. unpow2 74.2%

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

      3. distribute-rgt-out 79.4%

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

   8. Simplified 79.4%

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

   9. Taylor expanded in M around 0 79.4%

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

   10. Taylor expanded in n around inf 58.2%

       \[\leadsto 1 \cdot e^{\color{blue}{-1 \cdot \left(n \cdot \left(-0.5 \cdot M + 0.5 \cdot \left(m - M\right)\right)\right) + -0.25 \cdot {n}^{2}}} \]

   if -2.2000000000000001e-127 < n < 1.46e10

   1. Initial program 82.7%

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

   3. Taylor expanded in K around 0 92.1%

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

      1. cos-neg 92.1%

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

      2. sub-neg 92.1%

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

      3. sub-neg 92.1%

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

      4. associate--r+ 92.1%

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

      5. *-commutative 92.1%

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

      6. associate--r+ 92.1%

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

   5. Simplified 92.1%

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

   6. Taylor expanded in m around 0 68.7%

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

      1. +-commutative 68.7%

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

      2. unpow2 68.7%

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

      3. distribute-rgt-out 70.8%

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

   8. Simplified 70.8%

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

   9. Taylor expanded in M around 0 69.7%

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

   if 1.46e10 < n

   1. Initial program 70.0%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. associate--r+ 100.0%

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

      5. *-commutative 100.0%

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

      6. associate--r+ 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in m around 0 75.2%

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

      1. +-commutative 75.2%

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

      2. unpow2 75.2%

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

      3. distribute-rgt-out 83.6%

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

   8. Simplified 83.6%

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

   9. Taylor expanded in M around 0 83.6%

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

   10. Taylor expanded in n around inf 98.4%

       \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   11. Step-by-step derivation

       1. *-commutative 98.4%

          \[\leadsto 1 \cdot e^{\color{blue}{{n}^{2} \cdot -0.25}} \]

   12. Simplified 98.4%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.2 \cdot 10^{-127}:\\ \;\;\;\;e^{n \cdot \left(0.5 \cdot \left(M - m\right) - M \cdot -0.5\right) + {n}^{2} \cdot -0.25}\\ \mathbf{elif}\;n \leq 14600000000:\\ \;\;\;\;e^{\left|m - n\right| + \left(\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{{n}^{2} \cdot -0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -2.12 \cdot 10^{-13} \lor \neg \left(M \leq 5.3 \cdot 10^{+14}\right):\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - \left(n \cdot 0.5\right) \cdot \left(m + n \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -2.12e-13) (not (<= M 5.3e+14)))
   (exp (- (pow M 2.0)))
   (exp (- (- (fabs (- m n)) l) (* (* n 0.5) (+ m (* n 0.5)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -2.12e-13) || !(M <= 5.3e+14)) {
		tmp = exp(-pow(M, 2.0));
	} else {
		tmp = exp(((fabs((m - n)) - l) - ((n * 0.5) * (m + (n * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((m_1 <= (-2.12d-13)) .or. (.not. (m_1 <= 5.3d+14))) then
        tmp = exp(-(m_1 ** 2.0d0))
    else
        tmp = exp(((abs((m - n)) - l) - ((n * 0.5d0) * (m + (n * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -2.12e-13) || !(M <= 5.3e+14)) {
		tmp = Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.exp(((Math.abs((m - n)) - l) - ((n * 0.5) * (m + (n * 0.5)))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -2.12e-13) or not (M <= 5.3e+14):
		tmp = math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.exp(((math.fabs((m - n)) - l) - ((n * 0.5) * (m + (n * 0.5)))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -2.12e-13) || !(M <= 5.3e+14))
		tmp = exp(Float64(-(M ^ 2.0)));
	else
		tmp = exp(Float64(Float64(abs(Float64(m - n)) - l) - Float64(Float64(n * 0.5) * Float64(m + Float64(n * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -2.12e-13) || ~((M <= 5.3e+14)))
		tmp = exp(-(M ^ 2.0));
	else
		tmp = exp(((abs((m - n)) - l) - ((n * 0.5) * (m + (n * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -2.12e-13], N[Not[LessEqual[M, 5.3e+14]], $MachinePrecision]], N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision], N[Exp[N[(N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[(N[(n * 0.5), $MachinePrecision] * N[(m + N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -2.1200000000000001e-13 or 5.3e14 < M

   1. Initial program 75.2%

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

   3. Taylor expanded in K around 0 98.6%

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

      1. cos-neg 98.6%

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

      2. sub-neg 98.6%

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

      3. sub-neg 98.6%

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

      4. associate--r+ 98.6%

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

      5. *-commutative 98.6%

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

      6. associate--r+ 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in m around 0 81.5%

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

      1. +-commutative 81.5%

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

      2. unpow2 81.5%

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

      3. distribute-rgt-out 87.8%

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

   8. Simplified 87.8%

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

   9. Taylor expanded in M around 0 87.1%

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

   10. Taylor expanded in M around inf 95.9%

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

       1. mul-1-neg 95.9%

          \[\leadsto 1 \cdot e^{\color{blue}{-{M}^{2}}} \]

   12. Simplified 95.9%

       \[\leadsto 1 \cdot e^{\color{blue}{-{M}^{2}}} \]

   if -2.1200000000000001e-13 < M < 5.3e14

   1. Initial program 76.3%

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

   3. Taylor expanded in K around 0 90.0%

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

      1. cos-neg 90.0%

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

      2. sub-neg 90.0%

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

      3. sub-neg 90.0%

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

      4. associate--r+ 90.0%

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

      5. *-commutative 90.0%

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

      6. associate--r+ 90.0%

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

   5. Simplified 90.0%

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

   6. Taylor expanded in m around 0 60.4%

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

      1. +-commutative 60.4%

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

      2. unpow2 60.4%

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

      3. distribute-rgt-out 63.2%

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

   8. Simplified 63.2%

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

   9. Taylor expanded in M around 0 63.2%

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

   10. Taylor expanded in M around 0 62.3%

       \[\leadsto 1 \cdot e^{\color{blue}{\left|m - n\right| - \left(\ell + 0.5 \cdot \left(n \cdot \left(m + 0.5 \cdot n\right)\right)\right)}} \]
   11. Step-by-step derivation

       1. associate--r+ 62.3%

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

       2. associate-*r* 62.3%

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

       3. *-commutative 62.3%

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

       4. *-commutative 62.3%

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

   12. Simplified 62.3%

       \[\leadsto 1 \cdot e^{\color{blue}{\left(\left|m - n\right| - \ell\right) - \left(n \cdot 0.5\right) \cdot \left(m + n \cdot 0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2.12 \cdot 10^{-13} \lor \neg \left(M \leq 5.3 \cdot 10^{+14}\right):\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - \left(n \cdot 0.5\right) \cdot \left(m + n \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7.6 \cdot 10^{+71}:\\ \;\;\;\;e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{elif}\;n \leq 55:\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{{n}^{2} \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n -7.6e+71)
   (exp (* m (- M (* n 0.5))))
   (if (<= n 55.0) (exp (- (pow M 2.0))) (exp (* (pow n 2.0) -0.25)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= -7.6e+71) {
		tmp = exp((m * (M - (n * 0.5))));
	} else if (n <= 55.0) {
		tmp = exp(-pow(M, 2.0));
	} else {
		tmp = exp((pow(n, 2.0) * -0.25));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= (-7.6d+71)) then
        tmp = exp((m * (m_1 - (n * 0.5d0))))
    else if (n <= 55.0d0) then
        tmp = exp(-(m_1 ** 2.0d0))
    else
        tmp = exp(((n ** 2.0d0) * (-0.25d0)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= -7.6e+71) {
		tmp = Math.exp((m * (M - (n * 0.5))));
	} else if (n <= 55.0) {
		tmp = Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.exp((Math.pow(n, 2.0) * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= -7.6e+71:
		tmp = math.exp((m * (M - (n * 0.5))))
	elif n <= 55.0:
		tmp = math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.exp((math.pow(n, 2.0) * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= -7.6e+71)
		tmp = exp(Float64(m * Float64(M - Float64(n * 0.5))));
	elseif (n <= 55.0)
		tmp = exp(Float64(-(M ^ 2.0)));
	else
		tmp = exp(Float64((n ^ 2.0) * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= -7.6e+71)
		tmp = exp((m * (M - (n * 0.5))));
	elseif (n <= 55.0)
		tmp = exp(-(M ^ 2.0));
	else
		tmp = exp(((n ^ 2.0) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, -7.6e+71], N[Exp[N[(m * N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 55.0], N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision], N[Exp[N[(N[Power[n, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.6000000000000001e71

   1. Initial program 66.1%

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

   3. Taylor expanded in K around 0 98.2%

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

      1. cos-neg 98.2%

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

      2. sub-neg 98.2%

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

      3. sub-neg 98.2%

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

      4. associate--r+ 98.2%

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

      5. *-commutative 98.2%

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

      6. associate--r+ 98.2%

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

   5. Simplified 98.2%

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

   6. Taylor expanded in m around 0 80.4%

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

      1. +-commutative 80.4%

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

      2. unpow2 80.4%

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

      3. distribute-rgt-out 85.9%

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

   8. Simplified 85.9%

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

   9. Taylor expanded in M around 0 85.9%

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

   10. Taylor expanded in m around inf 47.5%

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

   if -7.6000000000000001e71 < n < 55

   1. Initial program 81.5%

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

   3. Taylor expanded in K around 0 91.2%

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

      1. cos-neg 91.2%

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

      2. sub-neg 91.2%

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

      3. sub-neg 91.2%

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

      4. associate--r+ 91.2%

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

      5. *-commutative 91.2%

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

      6. associate--r+ 91.2%

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

   5. Simplified 91.2%

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

   6. Taylor expanded in m around 0 67.2%

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

      1. +-commutative 67.2%

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

      2. unpow2 67.2%

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

      3. distribute-rgt-out 70.1%

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

   8. Simplified 70.1%

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

   9. Taylor expanded in M around 0 69.4%

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

   10. Taylor expanded in M around inf 63.0%

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

       1. mul-1-neg 63.0%

          \[\leadsto 1 \cdot e^{\color{blue}{-{M}^{2}}} \]

   12. Simplified 63.0%

       \[\leadsto 1 \cdot e^{\color{blue}{-{M}^{2}}} \]

   if 55 < n

   1. Initial program 71.4%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. associate--r+ 100.0%

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

      5. *-commutative 100.0%

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

      6. associate--r+ 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in m around 0 76.4%

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

      1. +-commutative 76.4%

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

      2. unpow2 76.4%

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

      3. distribute-rgt-out 84.4%

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

   8. Simplified 84.4%

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

   9. Taylor expanded in M around 0 84.4%

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

   10. Taylor expanded in n around inf 98.4%

       \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   11. Step-by-step derivation

       1. *-commutative 98.4%

          \[\leadsto 1 \cdot e^{\color{blue}{{n}^{2} \cdot -0.25}} \]

   12. Simplified 98.4%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7.6 \cdot 10^{+71}:\\ \;\;\;\;e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{elif}\;n \leq 55:\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{{n}^{2} \cdot -0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3400000000000:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 0.0065:\\ \;\;\;\;e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -3400000000000.0)
   (* (cos M) (exp l))
   (if (<= l 0.0065) (exp (* m (- M (* n 0.5)))) (/ (cos M) (exp l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -3400000000000.0) {
		tmp = cos(M) * exp(l);
	} else if (l <= 0.0065) {
		tmp = exp((m * (M - (n * 0.5))));
	} 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 (l <= (-3400000000000.0d0)) then
        tmp = cos(m_1) * exp(l)
    else if (l <= 0.0065d0) then
        tmp = exp((m * (m_1 - (n * 0.5d0))))
    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 (l <= -3400000000000.0) {
		tmp = Math.cos(M) * Math.exp(l);
	} else if (l <= 0.0065) {
		tmp = Math.exp((m * (M - (n * 0.5))));
	} else {
		tmp = Math.cos(M) / Math.exp(l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -3400000000000.0:
		tmp = math.cos(M) * math.exp(l)
	elif l <= 0.0065:
		tmp = math.exp((m * (M - (n * 0.5))))
	else:
		tmp = math.cos(M) / math.exp(l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -3400000000000.0)
		tmp = Float64(cos(M) * exp(l));
	elseif (l <= 0.0065)
		tmp = exp(Float64(m * Float64(M - Float64(n * 0.5))));
	else
		tmp = Float64(cos(M) / exp(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -3400000000000.0)
		tmp = cos(M) * exp(l);
	elseif (l <= 0.0065)
		tmp = exp((m * (M - (n * 0.5))));
	else
		tmp = cos(M) / exp(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -3400000000000.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.0065], N[Exp[N[(m * N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.4e12

   1. Initial program 81.3%

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

   3. Taylor expanded in K around 0 91.3%

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

      1. cos-neg 91.3%

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

      2. sub-neg 91.3%

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

      3. sub-neg 91.3%

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

      4. associate--r+ 91.3%

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

      5. *-commutative 91.3%

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

      6. associate--r+ 91.3%

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

   5. Simplified 91.3%

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

   6. Taylor expanded in m around 0 67.8%

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

      1. +-commutative 67.8%

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

      2. unpow2 67.8%

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

      3. distribute-rgt-out 71.6%

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

   8. Simplified 71.6%

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

   9. Taylor expanded in l around inf 17.4%

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

       1. mul-1-neg 17.4%

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

   11. Simplified 17.4%

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

       1. pow1 17.4%

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

       2. add-sqr-sqrt 17.4%

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

       3. sqrt-unprod 17.4%

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

       4. sqr-neg 17.4%

          \[\leadsto {\left(\cos M \cdot e^{\sqrt{\color{blue}{\ell \cdot \ell}}}\right)}^{1} \]

       5. sqrt-unprod 0.0%

          \[\leadsto {\left(\cos M \cdot e^{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{1} \]

       6. add-sqr-sqrt 75.4%

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

   13. Applied egg-rr 75.4%

       \[\leadsto \color{blue}{{\left(\cos M \cdot e^{\ell}\right)}^{1}} \]
   14. Step-by-step derivation

       1. unpow1 75.4%

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

   15. Simplified 75.4%

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

   if -3.4e12 < l < 0.0064999999999999997

   1. Initial program 67.8%

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

   3. Taylor expanded in K around 0 94.9%

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

      1. cos-neg 94.9%

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

      2. sub-neg 94.9%

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

      3. sub-neg 94.9%

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

      4. associate--r+ 94.9%

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

      5. *-commutative 94.9%

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

      6. associate--r+ 94.9%

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

   5. Simplified 94.9%

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

   6. Taylor expanded in m around 0 70.5%

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

      1. +-commutative 70.5%

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

      2. unpow2 70.5%

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

      3. distribute-rgt-out 75.6%

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

   8. Simplified 75.6%

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

   9. Taylor expanded in M around 0 75.6%

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

   10. Taylor expanded in m around inf 44.9%

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

   if 0.0064999999999999997 < l

   1. Initial program 84.2%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. associate--r+ 100.0%

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

      5. *-commutative 100.0%

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

      6. associate--r+ 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in m around 0 82.6%

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

      1. +-commutative 82.6%

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

      2. unpow2 82.6%

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

      3. distribute-rgt-out 87.9%

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

   8. Simplified 87.9%

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

   9. Taylor expanded in l around inf 98.3%

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

       1. mul-1-neg 98.3%

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

   11. Simplified 98.3%

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

       1. exp-neg 98.3%

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

       2. un-div-inv 98.3%

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

   13. Applied egg-rr 98.3%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3400000000000:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 0.0065:\\ \;\;\;\;e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -950000:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 720:\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -950000.0)
   (* (cos M) (exp l))
   (if (<= l 720.0) (exp (- (pow M 2.0))) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -950000.0) {
		tmp = cos(M) * exp(l);
	} else if (l <= 720.0) {
		tmp = exp(-pow(M, 2.0));
	} 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 <= (-950000.0d0)) then
        tmp = cos(m_1) * exp(l)
    else if (l <= 720.0d0) then
        tmp = exp(-(m_1 ** 2.0d0))
    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 <= -950000.0) {
		tmp = Math.cos(M) * Math.exp(l);
	} else if (l <= 720.0) {
		tmp = Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -950000.0:
		tmp = math.cos(M) * math.exp(l)
	elif l <= 720.0:
		tmp = math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -950000.0)
		tmp = Float64(cos(M) * exp(l));
	elseif (l <= 720.0)
		tmp = exp(Float64(-(M ^ 2.0)));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -950000.0)
		tmp = cos(M) * exp(l);
	elseif (l <= 720.0)
		tmp = exp(-(M ^ 2.0));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -950000.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 720.0], N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.5e5

   1. Initial program 80.2%

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

   3. Taylor expanded in K around 0 91.4%

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

      1. cos-neg 91.4%

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

      2. sub-neg 91.4%

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

      3. sub-neg 91.4%

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

      4. associate--r+ 91.4%

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

      5. *-commutative 91.4%

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

      6. associate--r+ 91.4%

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

   5. Simplified 91.4%

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

   6. Taylor expanded in m around 0 68.2%

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

      1. +-commutative 68.2%

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

      2. unpow2 68.2%

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

      3. distribute-rgt-out 71.9%

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

   8. Simplified 71.9%

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

   9. Taylor expanded in l around inf 17.2%

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

       1. mul-1-neg 17.2%

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

   11. Simplified 17.2%

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

       1. pow1 17.2%

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

       2. add-sqr-sqrt 17.2%

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

       3. sqrt-unprod 17.2%

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

       4. sqr-neg 17.2%

          \[\leadsto {\left(\cos M \cdot e^{\sqrt{\color{blue}{\ell \cdot \ell}}}\right)}^{1} \]

       5. sqrt-unprod 0.0%

          \[\leadsto {\left(\cos M \cdot e^{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{1} \]

       6. add-sqr-sqrt 75.7%

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

   13. Applied egg-rr 75.7%

       \[\leadsto \color{blue}{{\left(\cos M \cdot e^{\ell}\right)}^{1}} \]
   14. Step-by-step derivation

       1. unpow1 75.7%

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

   15. Simplified 75.7%

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

   if -9.5e5 < l < 720

   1. Initial program 67.8%

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

   3. Taylor expanded in K around 0 94.9%

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

      1. cos-neg 94.9%

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

      2. sub-neg 94.9%

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

      3. sub-neg 94.9%

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

      4. associate--r+ 94.9%

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

      5. *-commutative 94.9%

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

      6. associate--r+ 94.9%

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

   5. Simplified 94.9%

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

   6. Taylor expanded in m around 0 70.5%

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

      1. +-commutative 70.5%

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

      2. unpow2 70.5%

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

      3. distribute-rgt-out 75.6%

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

   8. Simplified 75.6%

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

   9. Taylor expanded in M around 0 75.6%

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

   10. Taylor expanded in M around inf 68.2%

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

       1. mul-1-neg 68.2%

          \[\leadsto 1 \cdot e^{\color{blue}{-{M}^{2}}} \]

   12. Simplified 68.2%

       \[\leadsto 1 \cdot e^{\color{blue}{-{M}^{2}}} \]

   if 720 < l

   1. Initial program 85.7%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. associate--r+ 100.0%

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

      5. *-commutative 100.0%

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

      6. associate--r+ 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in m around 0 82.3%

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

      1. +-commutative 82.3%

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

      2. unpow2 82.3%

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

      3. distribute-rgt-out 87.7%

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

   8. Simplified 87.7%

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

   9. Taylor expanded in l around inf 100.0%

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

       1. mul-1-neg 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in M around 0 100.0%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -950000:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 720:\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3400000000000:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 0.0065:\\ \;\;\;\;e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -3400000000000.0)
   (* (cos M) (exp l))
   (if (<= l 0.0065) (exp (* m (- M (* n 0.5)))) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -3400000000000.0) {
		tmp = cos(M) * exp(l);
	} else if (l <= 0.0065) {
		tmp = exp((m * (M - (n * 0.5))));
	} 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 <= (-3400000000000.0d0)) then
        tmp = cos(m_1) * exp(l)
    else if (l <= 0.0065d0) then
        tmp = exp((m * (m_1 - (n * 0.5d0))))
    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 <= -3400000000000.0) {
		tmp = Math.cos(M) * Math.exp(l);
	} else if (l <= 0.0065) {
		tmp = Math.exp((m * (M - (n * 0.5))));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -3400000000000.0:
		tmp = math.cos(M) * math.exp(l)
	elif l <= 0.0065:
		tmp = math.exp((m * (M - (n * 0.5))))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -3400000000000.0)
		tmp = Float64(cos(M) * exp(l));
	elseif (l <= 0.0065)
		tmp = exp(Float64(m * Float64(M - Float64(n * 0.5))));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -3400000000000.0)
		tmp = cos(M) * exp(l);
	elseif (l <= 0.0065)
		tmp = exp((m * (M - (n * 0.5))));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -3400000000000.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.0065], N[Exp[N[(m * N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.4e12

   1. Initial program 81.3%

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

   3. Taylor expanded in K around 0 91.3%

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

      1. cos-neg 91.3%

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

      2. sub-neg 91.3%

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

      3. sub-neg 91.3%

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

      4. associate--r+ 91.3%

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

      5. *-commutative 91.3%

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

      6. associate--r+ 91.3%

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

   5. Simplified 91.3%

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

   6. Taylor expanded in m around 0 67.8%

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

      1. +-commutative 67.8%

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

      2. unpow2 67.8%

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

      3. distribute-rgt-out 71.6%

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

   8. Simplified 71.6%

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

   9. Taylor expanded in l around inf 17.4%

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

       1. mul-1-neg 17.4%

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

   11. Simplified 17.4%

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

       1. pow1 17.4%

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

       2. add-sqr-sqrt 17.4%

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

       3. sqrt-unprod 17.4%

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

       4. sqr-neg 17.4%

          \[\leadsto {\left(\cos M \cdot e^{\sqrt{\color{blue}{\ell \cdot \ell}}}\right)}^{1} \]

       5. sqrt-unprod 0.0%

          \[\leadsto {\left(\cos M \cdot e^{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{1} \]

       6. add-sqr-sqrt 75.4%

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

   13. Applied egg-rr 75.4%

       \[\leadsto \color{blue}{{\left(\cos M \cdot e^{\ell}\right)}^{1}} \]
   14. Step-by-step derivation

       1. unpow1 75.4%

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

   15. Simplified 75.4%

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

   if -3.4e12 < l < 0.0064999999999999997

   1. Initial program 67.8%

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

   3. Taylor expanded in K around 0 94.9%

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

      1. cos-neg 94.9%

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

      2. sub-neg 94.9%

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

      3. sub-neg 94.9%

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

      4. associate--r+ 94.9%

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

      5. *-commutative 94.9%

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

      6. associate--r+ 94.9%

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

   5. Simplified 94.9%

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

   6. Taylor expanded in m around 0 70.5%

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

      1. +-commutative 70.5%

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

      2. unpow2 70.5%

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

      3. distribute-rgt-out 75.6%

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

   8. Simplified 75.6%

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

   9. Taylor expanded in M around 0 75.6%

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

   10. Taylor expanded in m around inf 44.9%

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

   if 0.0064999999999999997 < l

   1. Initial program 84.2%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. associate--r+ 100.0%

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

      5. *-commutative 100.0%

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

      6. associate--r+ 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in m around 0 82.6%

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

      1. +-commutative 82.6%

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

      2. unpow2 82.6%

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

      3. distribute-rgt-out 87.9%

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

   8. Simplified 87.9%

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

   9. Taylor expanded in l around inf 98.3%

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

       1. mul-1-neg 98.3%

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

   11. Simplified 98.3%

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

   12. Taylor expanded in M around 0 98.3%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3400000000000:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 0.0065:\\ \;\;\;\;e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 0.0065:\\ \;\;\;\;e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 0.0065) (exp (* m (- M (* n 0.5)))) (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 0.0065) {
		tmp = exp((m * (M - (n * 0.5))));
	} 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 <= 0.0065d0) then
        tmp = exp((m * (m_1 - (n * 0.5d0))))
    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 <= 0.0065) {
		tmp = Math.exp((m * (M - (n * 0.5))));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 0.0065:
		tmp = math.exp((m * (M - (n * 0.5))))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 0.0065)
		tmp = exp(Float64(m * Float64(M - Float64(n * 0.5))));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 0.0065)
		tmp = exp((m * (M - (n * 0.5))));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 0.0065], N[Exp[N[(m * N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 0.0064999999999999997

   1. Initial program 73.2%

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

   3. Taylor expanded in K around 0 93.4%

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

      1. cos-neg 93.4%

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

      2. sub-neg 93.4%

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

      3. sub-neg 93.4%

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

      4. associate--r+ 93.4%

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

      5. *-commutative 93.4%

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

      6. associate--r+ 93.4%

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

   5. Simplified 93.4%

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

   6. Taylor expanded in m around 0 69.4%

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

      1. +-commutative 69.4%

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

      2. unpow2 69.4%

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

      3. distribute-rgt-out 74.0%

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

   8. Simplified 74.0%

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

   9. Taylor expanded in M around 0 73.5%

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

   10. Taylor expanded in m around inf 41.4%

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

   if 0.0064999999999999997 < l

   1. Initial program 84.2%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. associate--r+ 100.0%

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

      5. *-commutative 100.0%

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

      6. associate--r+ 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in m around 0 82.6%

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

      1. +-commutative 82.6%

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

      2. unpow2 82.6%

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

      3. distribute-rgt-out 87.9%

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

   8. Simplified 87.9%

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

   9. Taylor expanded in l around inf 98.3%

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

       1. mul-1-neg 98.3%

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

   11. Simplified 98.3%

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

   12. Taylor expanded in M around 0 98.3%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 0.0065:\\ \;\;\;\;e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.3% 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 75.7%

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

3. Taylor expanded in K around 0 94.9%

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

   1. cos-neg 94.9%

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

   2. sub-neg 94.9%

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

   3. sub-neg 94.9%

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

   4. associate--r+ 94.9%

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

   5. *-commutative 94.9%

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

   6. associate--r+ 94.9%

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

5. Simplified 94.9%

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

6. Taylor expanded in m around 0 72.4%

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

   1. +-commutative 72.4%

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

   2. unpow2 72.4%

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

   3. distribute-rgt-out 77.1%

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

8. Simplified 77.1%

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

9. Taylor expanded in l around inf 32.2%

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

    1. mul-1-neg 32.2%

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

11. Simplified 32.2%

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

12. Taylor expanded in M around 0 31.8%

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

13. Final simplification 31.8%

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

Alternative 12: 6.9% 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 75.7%

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

3. Taylor expanded in K around 0 94.9%

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

   1. cos-neg 94.9%

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

   2. sub-neg 94.9%

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

   3. sub-neg 94.9%

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

   4. associate--r+ 94.9%

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

   5. *-commutative 94.9%

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

   6. associate--r+ 94.9%

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

5. Simplified 94.9%

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

6. Taylor expanded in m around 0 72.4%

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

   1. +-commutative 72.4%

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

   2. unpow2 72.4%

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

   3. distribute-rgt-out 77.1%

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

8. Simplified 77.1%

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

9. Taylor expanded in l around inf 32.2%

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

    1. mul-1-neg 32.2%

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

11. Simplified 32.2%

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

12. Taylor expanded in l around 0 6.4%

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

13. Final simplification 6.4%

    \[\leadsto \cos M \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(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)))))))