Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.9% → 96.5%

Time: 12.4s

Alternatives: 7

Speedup: 1.3×

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: 75.9% 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: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

Julia

function code(K, m, n, M, l)
	return Float64(cos(M) * exp(Float64(abs(Float64(n - m)) - Float64(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((n - m)) - (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[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(l + N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.9%

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

   1. +-commutative 72.9%

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

   2. +-commutative 72.9%

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

   3. fabs-sub 72.9%

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

   4. associate-/l* 73.5%

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

   5. +-commutative 73.5%

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

3. Simplified 73.5%

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

4. Taylor expanded in K around 0 96.6%

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

   1. cos-neg 96.6%

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

6. Simplified 96.6%

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

7. Final simplification 96.6%

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

Alternative 2: 85.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;m \leq -1.5 \cdot 10^{+19}:\\ \;\;\;\;\cos M \cdot e^{t_0 - \left(\left(m \cdot m\right) \cdot 0.25 + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t_0 + \left(\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) - \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<= m -1.5e+19)
     (* (cos M) (exp (- t_0 (+ (* (* m m) 0.25) l))))
     (*
      (cos M)
      (exp (+ t_0 (- (* (- (* n 0.5) M) (- (- M (* n 0.5)) m)) l)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if (m <= -1.5e+19) {
		tmp = cos(M) * exp((t_0 - (((m * m) * 0.25) + l)));
	} else {
		tmp = cos(M) * exp((t_0 + ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - 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) :: t_0
    real(8) :: tmp
    t_0 = abs((n - m))
    if (m <= (-1.5d+19)) then
        tmp = cos(m_1) * exp((t_0 - (((m * m) * 0.25d0) + l)))
    else
        tmp = cos(m_1) * exp((t_0 + ((((n * 0.5d0) - m_1) * ((m_1 - (n * 0.5d0)) - m)) - l)))
    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.abs((n - m));
	double tmp;
	if (m <= -1.5e+19) {
		tmp = Math.cos(M) * Math.exp((t_0 - (((m * m) * 0.25) + l)));
	} else {
		tmp = Math.cos(M) * Math.exp((t_0 + ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if m <= -1.5e+19:
		tmp = math.cos(M) * math.exp((t_0 - (((m * m) * 0.25) + l)))
	else:
		tmp = math.cos(M) * math.exp((t_0 + ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (m <= -1.5e+19)
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(Float64(Float64(m * m) * 0.25) + l))));
	else
		tmp = Float64(cos(M) * exp(Float64(t_0 + Float64(Float64(Float64(Float64(n * 0.5) - M) * Float64(Float64(M - Float64(n * 0.5)) - m)) - l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m));
	tmp = 0.0;
	if (m <= -1.5e+19)
		tmp = cos(M) * exp((t_0 - (((m * m) * 0.25) + l)));
	else
		tmp = cos(M) * exp((t_0 + ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[m, -1.5e+19], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(N[(N[(m * m), $MachinePrecision] * 0.25), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 + 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]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < -1.5e19

   1. Initial program 55.2%

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

      1. +-commutative 55.2%

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

      2. +-commutative 55.2%

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

      3. fabs-sub 55.2%

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

      4. associate-/l* 55.2%

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

      5. +-commutative 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in m around inf 88.1%

      \[\leadsto \cos M \cdot e^{\left(\left(-\color{blue}{0.25 \cdot {m}^{2}}\right) - \ell\right) + \left|n - m\right|} \]
   8. Step-by-step derivation

      1. *-commutative 88.1%

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

      2. unpow2 88.1%

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

   9. Simplified 88.1%

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

   if -1.5e19 < m

   1. Initial program 78.2%

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

      1. +-commutative 78.2%

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

      2. +-commutative 78.2%

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

      3. fabs-sub 78.2%

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

      4. associate-/l* 78.8%

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

      5. +-commutative 78.8%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in K around 0 95.7%

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

      1. cos-neg 95.7%

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

   6. Simplified 95.7%

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

   7. Taylor expanded in m around 0 78.8%

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

      1. +-commutative 78.8%

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

      2. unpow2 78.8%

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

      3. distribute-rgt-out 82.9%

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

      4. *-commutative 82.9%

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

      5. *-commutative 82.9%

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

   9. Simplified 82.9%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.5 \cdot 10^{+19}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - \left(\left(m \cdot m\right) \cdot 0.25 + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| + \left(\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) - \ell\right)}\\ \end{array} \]

Alternative 3: 69.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;m \leq -8.2 \cdot 10^{+14}:\\ \;\;\;\;\cos M \cdot e^{t_0 - \left(\left(m \cdot m\right) \cdot 0.25 + \ell\right)}\\ \mathbf{elif}\;m \leq 4.7 \cdot 10^{-278}:\\ \;\;\;\;\cos M \cdot e^{t_0 - \left(M \cdot M + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t_0 - \left(0.25 \cdot \left(n \cdot n\right) + \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<= m -8.2e+14)
     (* (cos M) (exp (- t_0 (+ (* (* m m) 0.25) l))))
     (if (<= m 4.7e-278)
       (* (cos M) (exp (- t_0 (+ (* M M) l))))
       (* (cos M) (exp (- t_0 (+ (* 0.25 (* n n)) l))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if (m <= -8.2e+14) {
		tmp = cos(M) * exp((t_0 - (((m * m) * 0.25) + l)));
	} else if (m <= 4.7e-278) {
		tmp = cos(M) * exp((t_0 - ((M * M) + l)));
	} else {
		tmp = cos(M) * exp((t_0 - ((0.25 * (n * n)) + 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) :: t_0
    real(8) :: tmp
    t_0 = abs((n - m))
    if (m <= (-8.2d+14)) then
        tmp = cos(m_1) * exp((t_0 - (((m * m) * 0.25d0) + l)))
    else if (m <= 4.7d-278) then
        tmp = cos(m_1) * exp((t_0 - ((m_1 * m_1) + l)))
    else
        tmp = cos(m_1) * exp((t_0 - ((0.25d0 * (n * n)) + l)))
    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.abs((n - m));
	double tmp;
	if (m <= -8.2e+14) {
		tmp = Math.cos(M) * Math.exp((t_0 - (((m * m) * 0.25) + l)));
	} else if (m <= 4.7e-278) {
		tmp = Math.cos(M) * Math.exp((t_0 - ((M * M) + l)));
	} else {
		tmp = Math.cos(M) * Math.exp((t_0 - ((0.25 * (n * n)) + l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if m <= -8.2e+14:
		tmp = math.cos(M) * math.exp((t_0 - (((m * m) * 0.25) + l)))
	elif m <= 4.7e-278:
		tmp = math.cos(M) * math.exp((t_0 - ((M * M) + l)))
	else:
		tmp = math.cos(M) * math.exp((t_0 - ((0.25 * (n * n)) + l)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (m <= -8.2e+14)
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(Float64(Float64(m * m) * 0.25) + l))));
	elseif (m <= 4.7e-278)
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(Float64(M * M) + l))));
	else
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(Float64(0.25 * Float64(n * n)) + l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m));
	tmp = 0.0;
	if (m <= -8.2e+14)
		tmp = cos(M) * exp((t_0 - (((m * m) * 0.25) + l)));
	elseif (m <= 4.7e-278)
		tmp = cos(M) * exp((t_0 - ((M * M) + l)));
	else
		tmp = cos(M) * exp((t_0 - ((0.25 * (n * n)) + l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[m, -8.2e+14], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(N[(N[(m * m), $MachinePrecision] * 0.25), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 4.7e-278], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(N[(M * M), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < -8.2e14

   1. Initial program 55.2%

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

      1. +-commutative 55.2%

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

      2. +-commutative 55.2%

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

      3. fabs-sub 55.2%

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

      4. associate-/l* 55.2%

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

      5. +-commutative 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in m around inf 88.1%

      \[\leadsto \cos M \cdot e^{\left(\left(-\color{blue}{0.25 \cdot {m}^{2}}\right) - \ell\right) + \left|n - m\right|} \]
   8. Step-by-step derivation

      1. *-commutative 88.1%

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

      2. unpow2 88.1%

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

   9. Simplified 88.1%

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

   if -8.2e14 < m < 4.6999999999999997e-278

   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. Step-by-step derivation

      1. +-commutative 82.7%

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

      2. +-commutative 82.7%

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

      3. fabs-sub 82.7%

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

      4. associate-/l* 84.3%

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

      5. +-commutative 84.3%

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

   3. Simplified 84.3%

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

   4. Taylor expanded in K around 0 94.8%

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

      1. cos-neg 94.8%

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

   6. Simplified 94.8%

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

   7. Taylor expanded in M around inf 65.1%

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

      1. unpow2 65.1%

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

   9. Simplified 65.1%

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

   if 4.6999999999999997e-278 < m

   1. Initial program 75.5%

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

      1. +-commutative 75.5%

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

      2. +-commutative 75.5%

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

      3. fabs-sub 75.5%

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

      4. associate-/l* 75.6%

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

      5. +-commutative 75.6%

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

   3. Simplified 75.6%

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

   4. Taylor expanded in K around 0 96.2%

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

      1. cos-neg 96.2%

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

   6. Simplified 96.2%

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

   7. Taylor expanded in n around inf 57.2%

      \[\leadsto \cos M \cdot e^{\left(\left(-\color{blue}{0.25 \cdot {n}^{2}}\right) - \ell\right) + \left|n - m\right|} \]
   8. Step-by-step derivation

      1. *-commutative 57.2%

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

      2. unpow2 57.2%

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

   9. Simplified 57.2%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -8.2 \cdot 10^{+14}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - \left(\left(m \cdot m\right) \cdot 0.25 + \ell\right)}\\ \mathbf{elif}\;m \leq 4.7 \cdot 10^{-278}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - \left(M \cdot M + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - \left(0.25 \cdot \left(n \cdot n\right) + \ell\right)}\\ \end{array} \]

Alternative 4: 67.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;m \leq -5000000000000:\\ \;\;\;\;\cos M \cdot e^{t_0 - \left(\left(m \cdot m\right) \cdot 0.25 + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t_0 - \left(M \cdot M + \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<= m -5000000000000.0)
     (* (cos M) (exp (- t_0 (+ (* (* m m) 0.25) l))))
     (* (cos M) (exp (- t_0 (+ (* M M) l)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if (m <= -5000000000000.0) {
		tmp = cos(M) * exp((t_0 - (((m * m) * 0.25) + l)));
	} else {
		tmp = cos(M) * exp((t_0 - ((M * M) + 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) :: t_0
    real(8) :: tmp
    t_0 = abs((n - m))
    if (m <= (-5000000000000.0d0)) then
        tmp = cos(m_1) * exp((t_0 - (((m * m) * 0.25d0) + l)))
    else
        tmp = cos(m_1) * exp((t_0 - ((m_1 * m_1) + l)))
    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.abs((n - m));
	double tmp;
	if (m <= -5000000000000.0) {
		tmp = Math.cos(M) * Math.exp((t_0 - (((m * m) * 0.25) + l)));
	} else {
		tmp = Math.cos(M) * Math.exp((t_0 - ((M * M) + l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if m <= -5000000000000.0:
		tmp = math.cos(M) * math.exp((t_0 - (((m * m) * 0.25) + l)))
	else:
		tmp = math.cos(M) * math.exp((t_0 - ((M * M) + l)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (m <= -5000000000000.0)
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(Float64(Float64(m * m) * 0.25) + l))));
	else
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(Float64(M * M) + l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m));
	tmp = 0.0;
	if (m <= -5000000000000.0)
		tmp = cos(M) * exp((t_0 - (((m * m) * 0.25) + l)));
	else
		tmp = cos(M) * exp((t_0 - ((M * M) + l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[m, -5000000000000.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(N[(N[(m * m), $MachinePrecision] * 0.25), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(N[(M * M), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < -5e12

   1. Initial program 55.2%

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

      1. +-commutative 55.2%

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

      2. +-commutative 55.2%

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

      3. fabs-sub 55.2%

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

      4. associate-/l* 55.2%

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

      5. +-commutative 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in m around inf 88.1%

      \[\leadsto \cos M \cdot e^{\left(\left(-\color{blue}{0.25 \cdot {m}^{2}}\right) - \ell\right) + \left|n - m\right|} \]
   8. Step-by-step derivation

      1. *-commutative 88.1%

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

      2. unpow2 88.1%

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

   9. Simplified 88.1%

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

   if -5e12 < m

   1. Initial program 78.2%

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

      1. +-commutative 78.2%

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

      2. +-commutative 78.2%

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

      3. fabs-sub 78.2%

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

      4. associate-/l* 78.8%

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

      5. +-commutative 78.8%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in K around 0 95.7%

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

      1. cos-neg 95.7%

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

   6. Simplified 95.7%

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

   7. Taylor expanded in M around inf 62.5%

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

      1. unpow2 62.5%

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

   9. Simplified 62.5%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5000000000000:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - \left(\left(m \cdot m\right) \cdot 0.25 + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - \left(M \cdot M + \ell\right)}\\ \end{array} \]

Alternative 5: 55.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|m - n\right|\\ \mathbf{if}\;M \leq -4.5 \cdot 10^{+37} \lor \neg \left(M \leq 2.9 \cdot 10^{+25}\right):\\ \;\;\;\;\cos M \cdot e^{t_0 - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t_0 - \ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- m n))))
   (if (or (<= M -4.5e+37) (not (<= M 2.9e+25)))
     (* (cos M) (exp (- t_0 (* M M))))
     (* (cos M) (exp (- t_0 l))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((m - n));
	double tmp;
	if ((M <= -4.5e+37) || !(M <= 2.9e+25)) {
		tmp = cos(M) * exp((t_0 - (M * M)));
	} else {
		tmp = cos(M) * exp((t_0 - l));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((m - n))
    if ((m_1 <= (-4.5d+37)) .or. (.not. (m_1 <= 2.9d+25))) then
        tmp = cos(m_1) * exp((t_0 - (m_1 * m_1)))
    else
        tmp = cos(m_1) * exp((t_0 - l))
    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.abs((m - n));
	double tmp;
	if ((M <= -4.5e+37) || !(M <= 2.9e+25)) {
		tmp = Math.cos(M) * Math.exp((t_0 - (M * M)));
	} else {
		tmp = Math.cos(M) * Math.exp((t_0 - l));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((m - n))
	tmp = 0
	if (M <= -4.5e+37) or not (M <= 2.9e+25):
		tmp = math.cos(M) * math.exp((t_0 - (M * M)))
	else:
		tmp = math.cos(M) * math.exp((t_0 - l))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(m - n))
	tmp = 0.0
	if ((M <= -4.5e+37) || !(M <= 2.9e+25))
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(M * M))));
	else
		tmp = Float64(cos(M) * exp(Float64(t_0 - l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((m - n));
	tmp = 0.0;
	if ((M <= -4.5e+37) || ~((M <= 2.9e+25)))
		tmp = cos(M) * exp((t_0 - (M * M)));
	else
		tmp = cos(M) * exp((t_0 - l));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[M, -4.5e+37], N[Not[LessEqual[M, 2.9e+25]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < -4.49999999999999962e37 or 2.8999999999999999e25 < M

   1. Initial program 80.8%

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

      1. +-commutative 80.8%

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

      2. +-commutative 80.8%

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

      3. fabs-sub 80.8%

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

      4. associate-/l* 80.8%

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

      5. +-commutative 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in K around 0 99.2%

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

      1. cos-neg 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in M around inf 88.5%

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

      1. unpow2 88.5%

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

   9. Simplified 88.5%

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

   10. Taylor expanded in M around inf 89.3%

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

       1. mul-1-neg 89.3%

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

       2. unpow2 89.3%

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

       3. distribute-rgt-neg-out 89.3%

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

   12. Simplified 89.3%

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

   if -4.49999999999999962e37 < M < 2.8999999999999999e25

   1. Initial program 66.0%

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

      1. +-commutative 66.0%

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

      2. +-commutative 66.0%

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

      3. fabs-sub 66.0%

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

      4. associate-/l* 67.0%

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

      5. +-commutative 67.0%

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

   3. Simplified 67.0%

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

   4. Taylor expanded in K around 0 94.4%

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

      1. cos-neg 94.4%

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

   6. Simplified 94.4%

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

   7. Taylor expanded in M around inf 30.2%

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

      1. unpow2 30.2%

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

   9. Simplified 30.2%

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

   10. Taylor expanded in M around 0 29.5%

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

       1. neg-mul-1 29.5%

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

   12. Simplified 29.5%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -4.5 \cdot 10^{+37} \lor \neg \left(M \leq 2.9 \cdot 10^{+25}\right):\\ \;\;\;\;\cos M \cdot e^{\left|m - n\right| - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|m - n\right| - \ell}\\ \end{array} \]

Alternative 6: 57.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

   1. +-commutative 72.9%

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

   2. +-commutative 72.9%

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

   3. fabs-sub 72.9%

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

   4. associate-/l* 73.5%

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

   5. +-commutative 73.5%

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

3. Simplified 73.5%

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

4. Taylor expanded in K around 0 96.6%

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

   1. cos-neg 96.6%

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

6. Simplified 96.6%

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

7. Taylor expanded in M around inf 57.6%

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

   1. unpow2 57.6%

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

9. Simplified 57.6%

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

10. Final simplification 57.6%

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

Alternative 7: 25.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return cos(M) * exp((fabs((m - n)) - 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 = cos(m_1) * exp((abs((m - n)) - l))
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));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

   1. +-commutative 72.9%

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

   2. +-commutative 72.9%

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

   3. fabs-sub 72.9%

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

   4. associate-/l* 73.5%

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

   5. +-commutative 73.5%

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

3. Simplified 73.5%

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

4. Taylor expanded in K around 0 96.6%

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

   1. cos-neg 96.6%

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

6. Simplified 96.6%

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

7. Taylor expanded in M around inf 57.6%

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

   1. unpow2 57.6%

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

9. Simplified 57.6%

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

10. Taylor expanded in M around 0 22.7%

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

    1. neg-mul-1 22.7%

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

12. Simplified 22.7%

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

13. Final simplification 22.7%

    \[\leadsto \cos M \cdot e^{\left|m - n\right| - \ell} \]

Reproduce

herbie shell --seed 2023272 
(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)))))))