Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.7% → 96.8%

Time: 15.0s

Alternatives: 10

Speedup: 2.0×

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: 76.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.8% accurate, 1.0× speedup

Math

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

FPCore

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

Fortran

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

Java

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.9%

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

3. Taylor expanded in K around 0 96.8%

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

   1. cos-neg 96.8%

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

5. Simplified 96.8%

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

6. Final simplification 96.8%

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

Alternative 2: 69.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -55:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq 5.4 \cdot 10^{-162}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - {M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -55.0)
   (exp (* -0.25 (pow m 2.0)))
   (if (<= m 5.4e-162)
     (exp (- (- (fabs (- m n)) l) (pow M 2.0)))
     (exp (* -0.25 (pow n 2.0))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -55.0) {
		tmp = exp((-0.25 * pow(m, 2.0)));
	} else if (m <= 5.4e-162) {
		tmp = exp(((fabs((m - n)) - l) - pow(M, 2.0)));
	} else {
		tmp = exp((-0.25 * pow(n, 2.0)));
	}
	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) :: tmp
    if (m <= (-55.0d0)) then
        tmp = exp(((-0.25d0) * (m ** 2.0d0)))
    else if (m <= 5.4d-162) then
        tmp = exp(((abs((m - n)) - l) - (m_1 ** 2.0d0)))
    else
        tmp = exp(((-0.25d0) * (n ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -55.0) {
		tmp = Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else if (m <= 5.4e-162) {
		tmp = Math.exp(((Math.abs((m - n)) - l) - Math.pow(M, 2.0)));
	} else {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -55.0:
		tmp = math.exp((-0.25 * math.pow(m, 2.0)))
	elif m <= 5.4e-162:
		tmp = math.exp(((math.fabs((m - n)) - l) - math.pow(M, 2.0)))
	else:
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -55.0)
		tmp = exp(Float64(-0.25 * (m ^ 2.0)));
	elseif (m <= 5.4e-162)
		tmp = exp(Float64(Float64(abs(Float64(m - n)) - l) - (M ^ 2.0)));
	else
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -55.0)
		tmp = exp((-0.25 * (m ^ 2.0)));
	elseif (m <= 5.4e-162)
		tmp = exp(((abs((m - n)) - l) - (M ^ 2.0)));
	else
		tmp = exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -55.0], N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, 5.4e-162], N[Exp[N[(N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -55

   1. Initial program 77.1%

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

   3. Taylor expanded in n around inf 87.1%

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

      1. *-commutative 87.1%

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

      2. associate-*l* 87.1%

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

   5. Simplified 87.1%

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

   6. Taylor expanded in K around 0 100.0%

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

   7. Taylor expanded in m around inf 97.2%

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

   if -55 < m < 5.39999999999999968e-162

   1. Initial program 86.3%

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

   3. Taylor expanded in n around inf 84.2%

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

      1. *-commutative 84.2%

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

      2. associate-*l* 84.2%

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

   5. Simplified 84.2%

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

   6. Taylor expanded in K around 0 93.6%

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

   7. Taylor expanded in M around inf 72.6%

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

   if 5.39999999999999968e-162 < m

   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 95.7%

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

      1. cos-neg 95.7%

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

   5. Simplified 95.7%

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

   6. Taylor expanded in n around inf 47.9%

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

   7. Taylor expanded in M around 0 47.8%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -55:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq 5.4 \cdot 10^{-162}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - {M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.9%

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

3. Taylor expanded in n around inf 85.5%

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

   1. *-commutative 85.5%

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

   2. associate-*l* 85.5%

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

5. Simplified 85.5%

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

6. Taylor expanded in K around 0 95.6%

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

7. Final simplification 95.6%

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

Alternative 4: 64.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-{M}^{2}}\\ \mathbf{if}\;m \leq -54:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq -4.6 \cdot 10^{-176}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq -1.3 \cdot 10^{-256}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{elif}\;m \leq 4.8 \cdot 10^{-162}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- (pow M 2.0)))))
   (if (<= m -54.0)
     (exp (* -0.25 (pow m 2.0)))
     (if (<= m -4.6e-176)
       t_0
       (if (<= m -1.3e-256)
         (/ (cos M) (exp l))
         (if (<= m 4.8e-162) t_0 (exp (* -0.25 (pow n 2.0)))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(-pow(M, 2.0));
	double tmp;
	if (m <= -54.0) {
		tmp = exp((-0.25 * pow(m, 2.0)));
	} else if (m <= -4.6e-176) {
		tmp = t_0;
	} else if (m <= -1.3e-256) {
		tmp = cos(M) / exp(l);
	} else if (m <= 4.8e-162) {
		tmp = t_0;
	} else {
		tmp = exp((-0.25 * pow(n, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-(m_1 ** 2.0d0))
    if (m <= (-54.0d0)) then
        tmp = exp(((-0.25d0) * (m ** 2.0d0)))
    else if (m <= (-4.6d-176)) then
        tmp = t_0
    else if (m <= (-1.3d-256)) then
        tmp = cos(m_1) / exp(l)
    else if (m <= 4.8d-162) then
        tmp = t_0
    else
        tmp = exp(((-0.25d0) * (n ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp(-Math.pow(M, 2.0));
	double tmp;
	if (m <= -54.0) {
		tmp = Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else if (m <= -4.6e-176) {
		tmp = t_0;
	} else if (m <= -1.3e-256) {
		tmp = Math.cos(M) / Math.exp(l);
	} else if (m <= 4.8e-162) {
		tmp = t_0;
	} else {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp(-math.pow(M, 2.0))
	tmp = 0
	if m <= -54.0:
		tmp = math.exp((-0.25 * math.pow(m, 2.0)))
	elif m <= -4.6e-176:
		tmp = t_0
	elif m <= -1.3e-256:
		tmp = math.cos(M) / math.exp(l)
	elif m <= 4.8e-162:
		tmp = t_0
	else:
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-(M ^ 2.0)))
	tmp = 0.0
	if (m <= -54.0)
		tmp = exp(Float64(-0.25 * (m ^ 2.0)));
	elseif (m <= -4.6e-176)
		tmp = t_0;
	elseif (m <= -1.3e-256)
		tmp = Float64(cos(M) / exp(l));
	elseif (m <= 4.8e-162)
		tmp = t_0;
	else
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(-(M ^ 2.0));
	tmp = 0.0;
	if (m <= -54.0)
		tmp = exp((-0.25 * (m ^ 2.0)));
	elseif (m <= -4.6e-176)
		tmp = t_0;
	elseif (m <= -1.3e-256)
		tmp = cos(M) / exp(l);
	elseif (m <= 4.8e-162)
		tmp = t_0;
	else
		tmp = exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]}, If[LessEqual[m, -54.0], N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, -4.6e-176], t$95$0, If[LessEqual[m, -1.3e-256], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 4.8e-162], t$95$0, N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if m < -54

   1. Initial program 77.1%

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

   3. Taylor expanded in n around inf 87.1%

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

      1. *-commutative 87.1%

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

      2. associate-*l* 87.1%

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

   5. Simplified 87.1%

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

   6. Taylor expanded in K around 0 100.0%

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

   7. Taylor expanded in m around inf 97.2%

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

   if -54 < m < -4.6000000000000003e-176 or -1.3e-256 < m < 4.8000000000000004e-162

   1. Initial program 86.3%

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

   3. Taylor expanded in n around inf 83.8%

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

      1. *-commutative 83.8%

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

      2. associate-*l* 83.8%

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

   5. Simplified 83.8%

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

   6. Taylor expanded in K around 0 92.5%

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

   7. Taylor expanded in M around inf 64.9%

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

      1. mul-1-neg 64.9%

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

   9. Simplified 64.9%

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

   if -4.6000000000000003e-176 < m < -1.3e-256

   1. Initial program 86.7%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in l around inf 60.8%

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

      1. mul-1-neg 60.8%

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

   8. Simplified 60.8%

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

      1. exp-neg 60.8%

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

      2. un-div-inv 60.8%

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

   10. Applied egg-rr 60.8%

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

   if 4.8000000000000004e-162 < m

   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 95.7%

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

      1. cos-neg 95.7%

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

   5. Simplified 95.7%

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

   6. Taylor expanded in n around inf 47.9%

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

   7. Taylor expanded in M around 0 47.8%

      \[\leadsto \color{blue}{e^{-0.25 \cdot {n}^{2}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -54:\\ \;\;\;\;e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq -4.6 \cdot 10^{-176}:\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{elif}\;m \leq -1.3 \cdot 10^{-256}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{elif}\;m \leq 4.8 \cdot 10^{-162}:\\ \;\;\;\;e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -0.00016 \lor \neg \left(n \leq 54\right):\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-{M}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= n -0.00016) (not (<= n 54.0)))
   (exp (* -0.25 (pow n 2.0)))
   (exp (- (pow M 2.0)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -0.00016) || !(n <= 54.0)) {
		tmp = exp((-0.25 * pow(n, 2.0)));
	} else {
		tmp = exp(-pow(M, 2.0));
	}
	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) :: tmp
    if ((n <= (-0.00016d0)) .or. (.not. (n <= 54.0d0))) then
        tmp = exp(((-0.25d0) * (n ** 2.0d0)))
    else
        tmp = exp(-(m_1 ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -0.00016) || !(n <= 54.0)) {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	} else {
		tmp = Math.exp(-Math.pow(M, 2.0));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (n <= -0.00016) or not (n <= 54.0):
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	else:
		tmp = math.exp(-math.pow(M, 2.0))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((n <= -0.00016) || !(n <= 54.0))
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	else
		tmp = exp(Float64(-(M ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((n <= -0.00016) || ~((n <= 54.0)))
		tmp = exp((-0.25 * (n ^ 2.0)));
	else
		tmp = exp(-(M ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[n, -0.00016], N[Not[LessEqual[n, 54.0]], $MachinePrecision]], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.60000000000000013e-4 or 54 < n

   1. Initial program 77.0%

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

   3. Taylor expanded in K around 0 99.2%

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

      1. cos-neg 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in n around inf 94.4%

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

   7. Taylor expanded in M around 0 94.4%

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

   if -1.60000000000000013e-4 < n < 54

   1. Initial program 84.4%

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

   3. Taylor expanded in n around inf 91.8%

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

      1. *-commutative 91.8%

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

      2. associate-*l* 91.8%

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

   5. Simplified 91.8%

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

   6. Taylor expanded in K around 0 93.2%

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

   7. Taylor expanded in M around inf 62.7%

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

      1. mul-1-neg 62.7%

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

   9. Simplified 62.7%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -0.00016 \lor \neg \left(n \leq 54\right):\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-{M}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.88:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{-5}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -0.88)
   (* (cos M) (exp l))
   (if (<= l 6.5e-5) (exp (* -0.25 (pow n 2.0))) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -0.88) {
		tmp = cos(M) * exp(l);
	} else if (l <= 6.5e-5) {
		tmp = exp((-0.25 * pow(n, 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 <= (-0.88d0)) then
        tmp = cos(m_1) * exp(l)
    else if (l <= 6.5d-5) then
        tmp = exp(((-0.25d0) * (n ** 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 <= -0.88) {
		tmp = Math.cos(M) * Math.exp(l);
	} else if (l <= 6.5e-5) {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -0.88:
		tmp = math.cos(M) * math.exp(l)
	elif l <= 6.5e-5:
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -0.88)
		tmp = Float64(cos(M) * exp(l));
	elseif (l <= 6.5e-5)
		tmp = exp(Float64(-0.25 * (n ^ 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 <= -0.88)
		tmp = cos(M) * exp(l);
	elseif (l <= 6.5e-5)
		tmp = exp((-0.25 * (n ^ 2.0)));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -0.88], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.5e-5], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -0.880000000000000004

   1. Initial program 86.8%

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

   3. Taylor expanded in K around 0 97.1%

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

      1. cos-neg 97.1%

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

   5. Simplified 97.1%

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

   6. Taylor expanded in l around inf 26.1%

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

      1. mul-1-neg 26.1%

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

   8. Simplified 26.1%

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

      1. pow1 26.1%

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

      2. add-sqr-sqrt 26.1%

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

      3. sqrt-unprod 26.1%

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

      4. sqr-neg 26.1%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 71.1%

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

   10. Applied egg-rr 71.1%

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

       1. unpow1 71.1%

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

   12. Simplified 71.1%

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

   if -0.880000000000000004 < l < 6.49999999999999943e-5

   1. Initial program 77.0%

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

   3. Taylor expanded in K around 0 95.3%

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

      1. cos-neg 95.3%

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

   5. Simplified 95.3%

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

   6. Taylor expanded in n around inf 56.5%

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

   7. Taylor expanded in M around 0 56.5%

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

   if 6.49999999999999943e-5 < l

   1. Initial program 82.8%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in l around inf 98.3%

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

      1. mul-1-neg 98.3%

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

   8. Simplified 98.3%

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

   9. Taylor expanded in M around 0 98.3%

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

Alternative 7: 48.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -8.8 \cdot 10^{-7}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -8.8e-7) (* (cos M) (exp l)) (/ (cos M) (exp l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -8.8e-7) {
		tmp = cos(M) * exp(l);
	} 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 <= (-8.8d-7)) then
        tmp = cos(m_1) * exp(l)
    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 <= -8.8e-7) {
		tmp = Math.cos(M) * Math.exp(l);
	} else {
		tmp = Math.cos(M) / Math.exp(l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -8.8e-7:
		tmp = math.cos(M) * math.exp(l)
	else:
		tmp = math.cos(M) / math.exp(l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -8.8e-7)
		tmp = Float64(cos(M) * exp(l));
	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 <= -8.8e-7)
		tmp = cos(M) * exp(l);
	else
		tmp = cos(M) / exp(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -8.8e-7], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -8.8000000000000004e-7

   1. Initial program 87.1%

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

   3. Taylor expanded in K around 0 97.1%

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

      1. cos-neg 97.1%

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

   5. Simplified 97.1%

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

   6. Taylor expanded in l around inf 25.5%

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

      1. mul-1-neg 25.5%

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

   8. Simplified 25.5%

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

      1. pow1 25.5%

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

      2. add-sqr-sqrt 25.5%

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

      3. sqrt-unprod 25.5%

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

      4. sqr-neg 25.5%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 69.1%

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

   10. Applied egg-rr 69.1%

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

       1. unpow1 69.1%

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

   12. Simplified 69.1%

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

   if -8.8000000000000004e-7 < l

   1. Initial program 78.6%

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

   3. Taylor expanded in K around 0 96.7%

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

      1. cos-neg 96.7%

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

   5. Simplified 96.7%

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

   6. Taylor expanded in l around inf 39.9%

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

      1. mul-1-neg 39.9%

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

   8. Simplified 39.9%

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

      1. exp-neg 39.9%

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

      2. un-div-inv 39.9%

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

   10. Applied egg-rr 39.9%

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

Alternative 8: 48.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 8.2 \cdot 10^{-8}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 8.2e-8) (* (cos M) (exp l)) (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 8.2e-8) {
		tmp = cos(M) * exp(l);
	} 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 <= 8.2d-8) then
        tmp = cos(m_1) * exp(l)
    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 <= 8.2e-8) {
		tmp = Math.cos(M) * Math.exp(l);
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 8.2e-8:
		tmp = math.cos(M) * math.exp(l)
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 8.2e-8)
		tmp = Float64(cos(M) * exp(l));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 8.2e-8)
		tmp = cos(M) * exp(l);
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 8.2e-8], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 8.20000000000000063e-8

   1. Initial program 80.4%

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

   3. Taylor expanded in K around 0 95.9%

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

      1. cos-neg 95.9%

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

   5. Simplified 95.9%

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

   6. Taylor expanded in l around inf 17.7%

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

      1. mul-1-neg 17.7%

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

   8. Simplified 17.7%

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

      1. pow1 17.7%

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

      2. add-sqr-sqrt 12.7%

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

      3. sqrt-unprod 17.7%

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

      4. sqr-neg 17.7%

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

      5. sqrt-unprod 5.0%

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

      6. add-sqr-sqrt 33.1%

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

   10. Applied egg-rr 33.1%

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

       1. unpow1 33.1%

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

   12. Simplified 33.1%

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

   if 8.20000000000000063e-8 < l

   1. Initial program 82.8%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in l around inf 98.3%

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

      1. mul-1-neg 98.3%

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

   8. Simplified 98.3%

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

   9. Taylor expanded in M around 0 98.3%

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

Alternative 9: 35.1% 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 80.9%

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

3. Taylor expanded in K around 0 96.8%

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

   1. cos-neg 96.8%

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

5. Simplified 96.8%

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

6. Taylor expanded in l around inf 35.9%

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

   1. mul-1-neg 35.9%

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

8. Simplified 35.9%

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

9. Taylor expanded in M around 0 34.8%

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

Alternative 10: 7.2% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.9%

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

3. Taylor expanded in K around 0 96.8%

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

   1. cos-neg 96.8%

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

5. Simplified 96.8%

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

6. Taylor expanded in l around inf 35.9%

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

   1. mul-1-neg 35.9%

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

8. Simplified 35.9%

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

9. Taylor expanded in l around 0 8.2%

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

Reproduce

herbie shell --seed 2024108 
(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)))))))