Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.9% → 96.5%

Time: 29.7s

Alternatives: 9

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

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

FPCore

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

C

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

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos(m_1) * exp((abs((n - m)) - (((((m + n) / 2.0d0) - m_1) ** 2.0d0) + 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)) - (Math.pow((((m + n) / 2.0) - M), 2.0) + l)));
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 75.3%

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

   1. +-commutative 75.3%

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

   2. +-commutative 75.3%

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

   3. fabs-sub 75.3%

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

   4. associate-/l* 75.4%

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

   5. +-commutative 75.4%

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

3. Simplified 75.4%

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

4. Taylor expanded in K around 0 95.5%

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

   1. cos-neg 95.5%

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

6. Simplified 95.5%

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

7. Final simplification 95.5%

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

Alternative 2: 91.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;M \leq -4 \cdot 10^{+126} \lor \neg \left(M \leq 1.55 \cdot 10^{+77}\right):\\ \;\;\;\;\cos M \cdot e^{t_0 - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{t_0 - \left(\ell + {\left(m + n\right)}^{2} \cdot 0.25\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (or (<= M -4e+126) (not (<= M 1.55e+77)))
     (* (cos M) (exp (- t_0 (* M M))))
     (exp (- t_0 (+ l (* (pow (+ m n) 2.0) 0.25)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if ((M <= -4e+126) || !(M <= 1.55e+77)) {
		tmp = cos(M) * exp((t_0 - (M * M)));
	} else {
		tmp = exp((t_0 - (l + (pow((m + n), 2.0) * 0.25))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((n - m))
    if ((m_1 <= (-4d+126)) .or. (.not. (m_1 <= 1.55d+77))) then
        tmp = cos(m_1) * exp((t_0 - (m_1 * m_1)))
    else
        tmp = exp((t_0 - (l + (((m + n) ** 2.0d0) * 0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((n - m));
	double tmp;
	if ((M <= -4e+126) || !(M <= 1.55e+77)) {
		tmp = Math.cos(M) * Math.exp((t_0 - (M * M)));
	} else {
		tmp = Math.exp((t_0 - (l + (Math.pow((m + n), 2.0) * 0.25))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if (M <= -4e+126) or not (M <= 1.55e+77):
		tmp = math.cos(M) * math.exp((t_0 - (M * M)))
	else:
		tmp = math.exp((t_0 - (l + (math.pow((m + n), 2.0) * 0.25))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if ((M <= -4e+126) || !(M <= 1.55e+77))
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(M * M))));
	else
		tmp = exp(Float64(t_0 - Float64(l + Float64((Float64(m + n) ^ 2.0) * 0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m));
	tmp = 0.0;
	if ((M <= -4e+126) || ~((M <= 1.55e+77)))
		tmp = cos(M) * exp((t_0 - (M * M)));
	else
		tmp = exp((t_0 - (l + (((m + n) ^ 2.0) * 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[M, -4e+126], N[Not[LessEqual[M, 1.55e+77]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(t$95$0 - N[(l + N[(N[Power[N[(m + n), $MachinePrecision], 2.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < -3.9999999999999997e126 or 1.54999999999999999e77 < M

   1. Initial program 78.8%

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

      1. +-commutative 78.8%

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

      2. +-commutative 78.8%

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

      3. fabs-sub 78.8%

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

      4. associate-/l* 80.0%

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

      5. +-commutative 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|n - m\right|} \]
   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 97.7%

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

      1. mul-1-neg 97.7%

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

      2. unpow2 97.7%

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

      3. distribute-rgt-neg-in 97.7%

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

   9. Simplified 97.7%

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

   if -3.9999999999999997e126 < M < 1.54999999999999999e77

   1. Initial program 73.5%

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

      1. +-commutative 73.5%

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

      2. +-commutative 73.5%

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

      3. fabs-sub 73.5%

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

      4. associate-/l* 73.1%

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

      5. +-commutative 73.1%

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

   3. Simplified 73.1%

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

   4. Taylor expanded in K around 0 84.8%

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

      1. cos-neg 84.8%

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

      2. associate-*r* 84.8%

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

   6. Simplified 84.8%

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

   7. Taylor expanded in M around 0 92.1%

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

      1. *-commutative 92.1%

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

      2. +-commutative 92.1%

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

   9. Simplified 92.1%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -4 \cdot 10^{+126} \lor \neg \left(M \leq 1.55 \cdot 10^{+77}\right):\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - \left(\ell + {\left(m + n\right)}^{2} \cdot 0.25\right)}\\ \end{array} \]

Alternative 3: 62.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{\left(m + n\right) \cdot K}{2} - M\right) \cdot e^{\left(-\ell\right) - M \cdot M}\\ t_1 := \left|n - m\right|\\ \mathbf{if}\;n \leq -1.75 \cdot 10^{-299}:\\ \;\;\;\;\cos M \cdot e^{t_1 + \left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;n \leq 8.5 \cdot 10^{-192}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 6.2 \cdot 10^{-120}:\\ \;\;\;\;e^{t_1 - 0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 1.32 \cdot 10^{-101}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{elif}\;n \leq 2.9 \cdot 10^{+35}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;e^{t_1 - 0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (cos (- (/ (* (+ m n) K) 2.0) M)) (exp (- (- l) (* M M)))))
        (t_1 (fabs (- n m))))
   (if (<= n -1.75e-299)
     (* (cos M) (exp (+ t_1 (* (* m m) -0.25))))
     (if (<= n 8.5e-192)
       t_0
       (if (<= n 6.2e-120)
         (exp (- t_1 (* 0.25 (* m m))))
         (if (<= n 1.32e-101)
           (/ (cos M) (exp l))
           (if (<= n 2.9e+35) t_0 (exp (- t_1 (* 0.25 (* n n)))))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(((((m + n) * K) / 2.0) - M)) * exp((-l - (M * M)));
	double t_1 = fabs((n - m));
	double tmp;
	if (n <= -1.75e-299) {
		tmp = cos(M) * exp((t_1 + ((m * m) * -0.25)));
	} else if (n <= 8.5e-192) {
		tmp = t_0;
	} else if (n <= 6.2e-120) {
		tmp = exp((t_1 - (0.25 * (m * m))));
	} else if (n <= 1.32e-101) {
		tmp = cos(M) / exp(l);
	} else if (n <= 2.9e+35) {
		tmp = t_0;
	} else {
		tmp = exp((t_1 - (0.25 * (n * n))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(((((m + n) * k) / 2.0d0) - m_1)) * exp((-l - (m_1 * m_1)))
    t_1 = abs((n - m))
    if (n <= (-1.75d-299)) then
        tmp = cos(m_1) * exp((t_1 + ((m * m) * (-0.25d0))))
    else if (n <= 8.5d-192) then
        tmp = t_0
    else if (n <= 6.2d-120) then
        tmp = exp((t_1 - (0.25d0 * (m * m))))
    else if (n <= 1.32d-101) then
        tmp = cos(m_1) / exp(l)
    else if (n <= 2.9d+35) then
        tmp = t_0
    else
        tmp = exp((t_1 - (0.25d0 * (n * n))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.cos(((((m + n) * K) / 2.0) - M)) * Math.exp((-l - (M * M)));
	double t_1 = Math.abs((n - m));
	double tmp;
	if (n <= -1.75e-299) {
		tmp = Math.cos(M) * Math.exp((t_1 + ((m * m) * -0.25)));
	} else if (n <= 8.5e-192) {
		tmp = t_0;
	} else if (n <= 6.2e-120) {
		tmp = Math.exp((t_1 - (0.25 * (m * m))));
	} else if (n <= 1.32e-101) {
		tmp = Math.cos(M) / Math.exp(l);
	} else if (n <= 2.9e+35) {
		tmp = t_0;
	} else {
		tmp = Math.exp((t_1 - (0.25 * (n * n))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos(((((m + n) * K) / 2.0) - M)) * math.exp((-l - (M * M)))
	t_1 = math.fabs((n - m))
	tmp = 0
	if n <= -1.75e-299:
		tmp = math.cos(M) * math.exp((t_1 + ((m * m) * -0.25)))
	elif n <= 8.5e-192:
		tmp = t_0
	elif n <= 6.2e-120:
		tmp = math.exp((t_1 - (0.25 * (m * m))))
	elif n <= 1.32e-101:
		tmp = math.cos(M) / math.exp(l)
	elif n <= 2.9e+35:
		tmp = t_0
	else:
		tmp = math.exp((t_1 - (0.25 * (n * n))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(cos(Float64(Float64(Float64(Float64(m + n) * K) / 2.0) - M)) * exp(Float64(Float64(-l) - Float64(M * M))))
	t_1 = abs(Float64(n - m))
	tmp = 0.0
	if (n <= -1.75e-299)
		tmp = Float64(cos(M) * exp(Float64(t_1 + Float64(Float64(m * m) * -0.25))));
	elseif (n <= 8.5e-192)
		tmp = t_0;
	elseif (n <= 6.2e-120)
		tmp = exp(Float64(t_1 - Float64(0.25 * Float64(m * m))));
	elseif (n <= 1.32e-101)
		tmp = Float64(cos(M) / exp(l));
	elseif (n <= 2.9e+35)
		tmp = t_0;
	else
		tmp = exp(Float64(t_1 - Float64(0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos(((((m + n) * K) / 2.0) - M)) * exp((-l - (M * M)));
	t_1 = abs((n - m));
	tmp = 0.0;
	if (n <= -1.75e-299)
		tmp = cos(M) * exp((t_1 + ((m * m) * -0.25)));
	elseif (n <= 8.5e-192)
		tmp = t_0;
	elseif (n <= 6.2e-120)
		tmp = exp((t_1 - (0.25 * (m * m))));
	elseif (n <= 1.32e-101)
		tmp = cos(M) / exp(l);
	elseif (n <= 2.9e+35)
		tmp = t_0;
	else
		tmp = exp((t_1 - (0.25 * (n * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Cos[N[(N[(N[(N[(m + n), $MachinePrecision] * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-l) - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -1.75e-299], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$1 + N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 8.5e-192], t$95$0, If[LessEqual[n, 6.2e-120], N[Exp[N[(t$95$1 - N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 1.32e-101], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.9e+35], t$95$0, N[Exp[N[(t$95$1 - N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -1.74999999999999995e-299

   1. Initial program 77.6%

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

      1. +-commutative 77.6%

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

      2. +-commutative 77.6%

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

      3. fabs-sub 77.6%

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

      4. associate-/l* 78.6%

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

      5. +-commutative 78.6%

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

   3. Simplified 78.6%

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

   4. Taylor expanded in K around 0 95.8%

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

      1. cos-neg 95.8%

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

   6. Simplified 95.8%

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

   7. Taylor expanded in m around inf 49.6%

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

      1. *-commutative 49.6%

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

      2. unpow2 49.6%

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

   9. Simplified 49.6%

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

   if -1.74999999999999995e-299 < n < 8.49999999999999985e-192 or 1.32e-101 < n < 2.89999999999999995e35

   1. Initial program 79.3%

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

   2. Taylor expanded in M around inf 58.3%

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

      1. unpow2 58.3%

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

   4. Simplified 58.3%

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

   5. Taylor expanded in l around inf 72.4%

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

   if 8.49999999999999985e-192 < n < 6.20000000000000038e-120

   1. Initial program 66.7%

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

      1. +-commutative 66.7%

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

      2. +-commutative 66.7%

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

      3. fabs-sub 66.7%

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

      4. associate-/l* 66.7%

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

      5. +-commutative 66.7%

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

   3. Simplified 66.7%

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

   4. Taylor expanded in K around 0 66.7%

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

      1. cos-neg 66.7%

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

      2. associate-*r* 66.7%

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

   6. Simplified 66.7%

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

   7. Taylor expanded in M around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. +-commutative 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in m around inf 83.9%

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

       1. *-commutative 83.9%

          \[\leadsto e^{\left|n - m\right| - \color{blue}{{m}^{2} \cdot 0.25}} \]

       2. unpow2 83.9%

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

   12. Simplified 83.9%

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

   if 6.20000000000000038e-120 < n < 1.32e-101

   1. Initial program 75.4%

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

      1. +-commutative 75.4%

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

      2. +-commutative 75.4%

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

      3. fabs-sub 75.4%

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

      4. associate-/l* 50.4%

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

      5. +-commutative 50.4%

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

   3. Simplified 50.4%

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

   4. Taylor expanded in l around inf 1.2%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell} + \left|n - m\right|} \]
   5. Step-by-step derivation

      1. neg-mul-1 1.2%

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

   6. Simplified 1.2%

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

   7. Taylor expanded in l around inf 1.6%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   8. Step-by-step derivation

      1. neg-mul-1 1.6%

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

   9. Simplified 1.6%

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

   10. Taylor expanded in m around 0 1.3%

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

   11. Taylor expanded in K around 0 31.6%

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

       1. exp-neg 31.6%

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

       2. associate-*r/ 31.6%

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

       3. *-rgt-identity 31.6%

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

       4. cos-neg 31.6%

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

   13. Simplified 31.6%

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

   if 2.89999999999999995e35 < n

   1. Initial program 68.2%

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

      1. +-commutative 68.2%

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

      2. +-commutative 68.2%

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

      3. fabs-sub 68.2%

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

      4. associate-/l* 68.2%

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

      5. +-commutative 68.2%

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

   3. Simplified 68.2%

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

   4. Taylor expanded in K around 0 81.8%

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

      1. cos-neg 81.8%

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

      2. associate-*r* 81.8%

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

   6. Simplified 81.8%

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

   7. Taylor expanded in M around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. +-commutative 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in n around inf 94.0%

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

       1. *-commutative 94.0%

          \[\leadsto e^{\left|n - m\right| - \color{blue}{{n}^{2} \cdot 0.25}} \]

       2. unpow2 94.0%

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

   12. Simplified 94.0%

       \[\leadsto e^{\left|n - m\right| - \color{blue}{\left(n \cdot n\right) \cdot 0.25}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.75 \cdot 10^{-299}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| + \left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;n \leq 8.5 \cdot 10^{-192}:\\ \;\;\;\;\cos \left(\frac{\left(m + n\right) \cdot K}{2} - M\right) \cdot e^{\left(-\ell\right) - M \cdot M}\\ \mathbf{elif}\;n \leq 6.2 \cdot 10^{-120}:\\ \;\;\;\;e^{\left|n - m\right| - 0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 1.32 \cdot 10^{-101}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{elif}\;n \leq 2.9 \cdot 10^{+35}:\\ \;\;\;\;\cos \left(\frac{\left(m + n\right) \cdot K}{2} - M\right) \cdot e^{\left(-\ell\right) - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - 0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]

Alternative 4: 62.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{\left(m + n\right) \cdot K}{2} - M\right) \cdot e^{\left(-\ell\right) - M \cdot M}\\ t_1 := \left|n - m\right|\\ t_2 := e^{t_1 - 0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{if}\;n \leq -1.46 \cdot 10^{-297}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-192}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 7.1 \cdot 10^{-122}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;n \leq 1.66 \cdot 10^{-102}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{+33}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;e^{t_1 - 0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (cos (- (/ (* (+ m n) K) 2.0) M)) (exp (- (- l) (* M M)))))
        (t_1 (fabs (- n m)))
        (t_2 (exp (- t_1 (* 0.25 (* m m))))))
   (if (<= n -1.46e-297)
     t_2
     (if (<= n 8e-192)
       t_0
       (if (<= n 7.1e-122)
         t_2
         (if (<= n 1.66e-102)
           (/ (cos M) (exp l))
           (if (<= n 5.5e+33) t_0 (exp (- t_1 (* 0.25 (* n n)))))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(((((m + n) * K) / 2.0) - M)) * exp((-l - (M * M)));
	double t_1 = fabs((n - m));
	double t_2 = exp((t_1 - (0.25 * (m * m))));
	double tmp;
	if (n <= -1.46e-297) {
		tmp = t_2;
	} else if (n <= 8e-192) {
		tmp = t_0;
	} else if (n <= 7.1e-122) {
		tmp = t_2;
	} else if (n <= 1.66e-102) {
		tmp = cos(M) / exp(l);
	} else if (n <= 5.5e+33) {
		tmp = t_0;
	} else {
		tmp = exp((t_1 - (0.25 * (n * n))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos(((((m + n) * k) / 2.0d0) - m_1)) * exp((-l - (m_1 * m_1)))
    t_1 = abs((n - m))
    t_2 = exp((t_1 - (0.25d0 * (m * m))))
    if (n <= (-1.46d-297)) then
        tmp = t_2
    else if (n <= 8d-192) then
        tmp = t_0
    else if (n <= 7.1d-122) then
        tmp = t_2
    else if (n <= 1.66d-102) then
        tmp = cos(m_1) / exp(l)
    else if (n <= 5.5d+33) then
        tmp = t_0
    else
        tmp = exp((t_1 - (0.25d0 * (n * n))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.cos(((((m + n) * K) / 2.0) - M)) * Math.exp((-l - (M * M)));
	double t_1 = Math.abs((n - m));
	double t_2 = Math.exp((t_1 - (0.25 * (m * m))));
	double tmp;
	if (n <= -1.46e-297) {
		tmp = t_2;
	} else if (n <= 8e-192) {
		tmp = t_0;
	} else if (n <= 7.1e-122) {
		tmp = t_2;
	} else if (n <= 1.66e-102) {
		tmp = Math.cos(M) / Math.exp(l);
	} else if (n <= 5.5e+33) {
		tmp = t_0;
	} else {
		tmp = Math.exp((t_1 - (0.25 * (n * n))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos(((((m + n) * K) / 2.0) - M)) * math.exp((-l - (M * M)))
	t_1 = math.fabs((n - m))
	t_2 = math.exp((t_1 - (0.25 * (m * m))))
	tmp = 0
	if n <= -1.46e-297:
		tmp = t_2
	elif n <= 8e-192:
		tmp = t_0
	elif n <= 7.1e-122:
		tmp = t_2
	elif n <= 1.66e-102:
		tmp = math.cos(M) / math.exp(l)
	elif n <= 5.5e+33:
		tmp = t_0
	else:
		tmp = math.exp((t_1 - (0.25 * (n * n))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(cos(Float64(Float64(Float64(Float64(m + n) * K) / 2.0) - M)) * exp(Float64(Float64(-l) - Float64(M * M))))
	t_1 = abs(Float64(n - m))
	t_2 = exp(Float64(t_1 - Float64(0.25 * Float64(m * m))))
	tmp = 0.0
	if (n <= -1.46e-297)
		tmp = t_2;
	elseif (n <= 8e-192)
		tmp = t_0;
	elseif (n <= 7.1e-122)
		tmp = t_2;
	elseif (n <= 1.66e-102)
		tmp = Float64(cos(M) / exp(l));
	elseif (n <= 5.5e+33)
		tmp = t_0;
	else
		tmp = exp(Float64(t_1 - Float64(0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos(((((m + n) * K) / 2.0) - M)) * exp((-l - (M * M)));
	t_1 = abs((n - m));
	t_2 = exp((t_1 - (0.25 * (m * m))));
	tmp = 0.0;
	if (n <= -1.46e-297)
		tmp = t_2;
	elseif (n <= 8e-192)
		tmp = t_0;
	elseif (n <= 7.1e-122)
		tmp = t_2;
	elseif (n <= 1.66e-102)
		tmp = cos(M) / exp(l);
	elseif (n <= 5.5e+33)
		tmp = t_0;
	else
		tmp = exp((t_1 - (0.25 * (n * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Cos[N[(N[(N[(N[(m + n), $MachinePrecision] * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-l) - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(t$95$1 - N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -1.46e-297], t$95$2, If[LessEqual[n, 8e-192], t$95$0, If[LessEqual[n, 7.1e-122], t$95$2, If[LessEqual[n, 1.66e-102], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.5e+33], t$95$0, N[Exp[N[(t$95$1 - N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.4600000000000001e-297 or 8.0000000000000008e-192 < n < 7.09999999999999955e-122

   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. Step-by-step derivation

      1. +-commutative 77.1%

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

      2. +-commutative 77.1%

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

      3. fabs-sub 77.1%

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

      4. associate-/l* 78.1%

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

      5. +-commutative 78.1%

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

   3. Simplified 78.1%

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

   4. Taylor expanded in K around 0 84.3%

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

      1. cos-neg 84.3%

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

      2. associate-*r* 85.0%

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

   6. Simplified 85.0%

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

   7. Taylor expanded in M around 0 87.7%

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

      1. *-commutative 87.7%

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

      2. +-commutative 87.7%

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

   9. Simplified 87.7%

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

   10. Taylor expanded in m around inf 51.2%

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

       1. *-commutative 51.2%

          \[\leadsto e^{\left|n - m\right| - \color{blue}{{m}^{2} \cdot 0.25}} \]

       2. unpow2 51.2%

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

   12. Simplified 51.2%

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

   if -1.4600000000000001e-297 < n < 8.0000000000000008e-192 or 1.66000000000000001e-102 < n < 5.5000000000000006e33

   1. Initial program 79.3%

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

   2. Taylor expanded in M around inf 58.3%

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

      1. unpow2 58.3%

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

   4. Simplified 58.3%

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

   5. Taylor expanded in l around inf 72.4%

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

   if 7.09999999999999955e-122 < n < 1.66000000000000001e-102

   1. Initial program 75.4%

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

      1. +-commutative 75.4%

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

      2. +-commutative 75.4%

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

      3. fabs-sub 75.4%

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

      4. associate-/l* 50.4%

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

      5. +-commutative 50.4%

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

   3. Simplified 50.4%

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

   4. Taylor expanded in l around inf 1.2%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell} + \left|n - m\right|} \]
   5. Step-by-step derivation

      1. neg-mul-1 1.2%

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

   6. Simplified 1.2%

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

   7. Taylor expanded in l around inf 1.6%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   8. Step-by-step derivation

      1. neg-mul-1 1.6%

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

   9. Simplified 1.6%

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

   10. Taylor expanded in m around 0 1.3%

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

   11. Taylor expanded in K around 0 31.6%

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

       1. exp-neg 31.6%

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

       2. associate-*r/ 31.6%

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

       3. *-rgt-identity 31.6%

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

       4. cos-neg 31.6%

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

   13. Simplified 31.6%

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

   if 5.5000000000000006e33 < n

   1. Initial program 68.2%

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

      1. +-commutative 68.2%

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

      2. +-commutative 68.2%

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

      3. fabs-sub 68.2%

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

      4. associate-/l* 68.2%

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

      5. +-commutative 68.2%

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

   3. Simplified 68.2%

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

   4. Taylor expanded in K around 0 81.8%

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

      1. cos-neg 81.8%

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

      2. associate-*r* 81.8%

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

   6. Simplified 81.8%

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

   7. Taylor expanded in M around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. +-commutative 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in n around inf 94.0%

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

       1. *-commutative 94.0%

          \[\leadsto e^{\left|n - m\right| - \color{blue}{{n}^{2} \cdot 0.25}} \]

       2. unpow2 94.0%

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

   12. Simplified 94.0%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.46 \cdot 10^{-297}:\\ \;\;\;\;e^{\left|n - m\right| - 0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 8 \cdot 10^{-192}:\\ \;\;\;\;\cos \left(\frac{\left(m + n\right) \cdot K}{2} - M\right) \cdot e^{\left(-\ell\right) - M \cdot M}\\ \mathbf{elif}\;n \leq 7.1 \cdot 10^{-122}:\\ \;\;\;\;e^{\left|n - m\right| - 0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 1.66 \cdot 10^{-102}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{+33}:\\ \;\;\;\;\cos \left(\frac{\left(m + n\right) \cdot K}{2} - M\right) \cdot e^{\left(-\ell\right) - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|n - m\right| - 0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]

Alternative 5: 58.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;m \leq -8.5 \cdot 10^{+38}:\\ \;\;\;\;e^{t_0 - 0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -6.6 \cdot 10^{-149} \lor \neg \left(m \leq -9.5 \cdot 10^{-194}\right):\\ \;\;\;\;e^{t_0 - 0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(m \cdot \left(K \cdot 0.5\right)\right) \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<= m -8.5e+38)
     (exp (- t_0 (* 0.25 (* m m))))
     (if (or (<= m -6.6e-149) (not (<= m -9.5e-194)))
       (exp (- t_0 (* 0.25 (* n n))))
       (* (cos (* m (* K 0.5))) (exp (- l)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if (m <= -8.5e+38) {
		tmp = exp((t_0 - (0.25 * (m * m))));
	} else if ((m <= -6.6e-149) || !(m <= -9.5e-194)) {
		tmp = exp((t_0 - (0.25 * (n * n))));
	} else {
		tmp = cos((m * (K * 0.5))) * 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) :: t_0
    real(8) :: tmp
    t_0 = abs((n - m))
    if (m <= (-8.5d+38)) then
        tmp = exp((t_0 - (0.25d0 * (m * m))))
    else if ((m <= (-6.6d-149)) .or. (.not. (m <= (-9.5d-194)))) then
        tmp = exp((t_0 - (0.25d0 * (n * n))))
    else
        tmp = cos((m * (k * 0.5d0))) * exp(-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 <= -8.5e+38) {
		tmp = Math.exp((t_0 - (0.25 * (m * m))));
	} else if ((m <= -6.6e-149) || !(m <= -9.5e-194)) {
		tmp = Math.exp((t_0 - (0.25 * (n * n))));
	} else {
		tmp = Math.cos((m * (K * 0.5))) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if m <= -8.5e+38:
		tmp = math.exp((t_0 - (0.25 * (m * m))))
	elif (m <= -6.6e-149) or not (m <= -9.5e-194):
		tmp = math.exp((t_0 - (0.25 * (n * n))))
	else:
		tmp = math.cos((m * (K * 0.5))) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (m <= -8.5e+38)
		tmp = exp(Float64(t_0 - Float64(0.25 * Float64(m * m))));
	elseif ((m <= -6.6e-149) || !(m <= -9.5e-194))
		tmp = exp(Float64(t_0 - Float64(0.25 * Float64(n * n))));
	else
		tmp = Float64(cos(Float64(m * Float64(K * 0.5))) * exp(Float64(-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 <= -8.5e+38)
		tmp = exp((t_0 - (0.25 * (m * m))));
	elseif ((m <= -6.6e-149) || ~((m <= -9.5e-194)))
		tmp = exp((t_0 - (0.25 * (n * n))));
	else
		tmp = cos((m * (K * 0.5))) * exp(-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, -8.5e+38], N[Exp[N[(t$95$0 - N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[m, -6.6e-149], N[Not[LessEqual[m, -9.5e-194]], $MachinePrecision]], N[Exp[N[(t$95$0 - N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[N[(m * N[(K * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < -8.4999999999999997e38

   1. Initial program 69.8%

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

      1. +-commutative 69.8%

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

      2. +-commutative 69.8%

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

      3. fabs-sub 69.8%

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

      4. associate-/l* 69.8%

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

      5. +-commutative 69.8%

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

   3. Simplified 69.8%

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

   4. Taylor expanded in K around 0 81.1%

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

      1. cos-neg 81.1%

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

      2. associate-*r* 81.1%

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

   6. Simplified 81.1%

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

   7. Taylor expanded in M around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. +-commutative 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in m around inf 98.1%

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

       1. *-commutative 98.1%

          \[\leadsto e^{\left|n - m\right| - \color{blue}{{m}^{2} \cdot 0.25}} \]

       2. unpow2 98.1%

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

   12. Simplified 98.1%

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

   if -8.4999999999999997e38 < m < -6.60000000000000034e-149 or -9.50000000000000009e-194 < m

   1. Initial program 76.2%

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

      1. +-commutative 76.2%

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

      2. +-commutative 76.2%

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

      3. fabs-sub 76.2%

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

      4. associate-/l* 76.3%

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

      5. +-commutative 76.3%

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

   3. Simplified 76.3%

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

   4. Taylor expanded in K around 0 83.2%

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

      1. cos-neg 83.2%

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

      2. associate-*r* 83.7%

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

   6. Simplified 83.7%

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

   7. Taylor expanded in M around 0 83.2%

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

      1. *-commutative 83.2%

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

      2. +-commutative 83.2%

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

   9. Simplified 83.2%

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

   10. Taylor expanded in n around inf 55.6%

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

       1. *-commutative 55.6%

          \[\leadsto e^{\left|n - m\right| - \color{blue}{{n}^{2} \cdot 0.25}} \]

       2. unpow2 55.6%

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

   12. Simplified 55.6%

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

   if -6.60000000000000034e-149 < m < -9.50000000000000009e-194

   1. Initial program 85.0%

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

      1. +-commutative 85.0%

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

      2. +-commutative 85.0%

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

      3. fabs-sub 85.0%

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

      4. associate-/l* 85.0%

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

      5. +-commutative 85.0%

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

   3. Simplified 85.0%

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

   4. Taylor expanded in l around inf 44.1%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell} + \left|n - m\right|} \]
   5. Step-by-step derivation

      1. neg-mul-1 44.1%

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

   6. Simplified 44.1%

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

   7. Taylor expanded in l around inf 44.3%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   8. Step-by-step derivation

      1. neg-mul-1 44.3%

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

   9. Simplified 44.3%

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

   10. Taylor expanded in m around inf 52.7%

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

       1. associate-*r* 52.7%

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

   12. Simplified 52.7%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -8.5 \cdot 10^{+38}:\\ \;\;\;\;e^{\left|n - m\right| - 0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -6.6 \cdot 10^{-149} \lor \neg \left(m \leq -9.5 \cdot 10^{-194}\right):\\ \;\;\;\;e^{\left|n - m\right| - 0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(m \cdot \left(K \cdot 0.5\right)\right) \cdot e^{-\ell}\\ \end{array} \]

Alternative 6: 62.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.8 \cdot 10^{+39} \lor \neg \left(m \leq 7.8 \cdot 10^{+88}\right):\\ \;\;\;\;e^{\left|n - m\right| - 0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= m -1.8e+39) (not (<= m 7.8e+88)))
   (exp (- (fabs (- n m)) (* 0.25 (* m m))))
   (/ (cos M) (exp l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((m <= -1.8e+39) || !(m <= 7.8e+88)) {
		tmp = exp((fabs((n - m)) - (0.25 * (m * m))));
	} 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 ((m <= (-1.8d+39)) .or. (.not. (m <= 7.8d+88))) then
        tmp = exp((abs((n - m)) - (0.25d0 * (m * m))))
    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 ((m <= -1.8e+39) || !(m <= 7.8e+88)) {
		tmp = Math.exp((Math.abs((n - m)) - (0.25 * (m * m))));
	} else {
		tmp = Math.cos(M) / Math.exp(l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (m <= -1.8e+39) or not (m <= 7.8e+88):
		tmp = math.exp((math.fabs((n - m)) - (0.25 * (m * m))))
	else:
		tmp = math.cos(M) / math.exp(l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((m <= -1.8e+39) || !(m <= 7.8e+88))
		tmp = exp(Float64(abs(Float64(n - m)) - Float64(0.25 * Float64(m * m))));
	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 ((m <= -1.8e+39) || ~((m <= 7.8e+88)))
		tmp = exp((abs((n - m)) - (0.25 * (m * m))));
	else
		tmp = cos(M) / exp(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[m, -1.8e+39], N[Not[LessEqual[m, 7.8e+88]], $MachinePrecision]], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.79999999999999992e39 or 7.8000000000000002e88 < m

   1. Initial program 68.1%

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

      1. +-commutative 68.1%

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

      2. +-commutative 68.1%

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

      3. fabs-sub 68.1%

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

      4. associate-/l* 68.1%

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

      5. +-commutative 68.1%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in K around 0 80.9%

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

      1. cos-neg 80.9%

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

      2. associate-*r* 80.9%

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

   6. Simplified 80.9%

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

   7. Taylor expanded in M around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. +-commutative 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in m around inf 99.0%

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

       1. *-commutative 99.0%

          \[\leadsto e^{\left|n - m\right| - \color{blue}{{m}^{2} \cdot 0.25}} \]

       2. unpow2 99.0%

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

   12. Simplified 99.0%

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

   if -1.79999999999999992e39 < m < 7.8000000000000002e88

   1. Initial program 79.4%

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

      1. +-commutative 79.4%

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

      2. +-commutative 79.4%

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

      3. fabs-sub 79.4%

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

      4. associate-/l* 79.6%

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

      5. +-commutative 79.6%

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

   3. Simplified 79.6%

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

   4. Taylor expanded in l around inf 30.1%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell} + \left|n - m\right|} \]
   5. Step-by-step derivation

      1. neg-mul-1 30.1%

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

   6. Simplified 30.1%

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

   7. Taylor expanded in l around inf 35.2%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   8. Step-by-step derivation

      1. neg-mul-1 35.2%

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

   9. Simplified 35.2%

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

   10. Taylor expanded in m around 0 35.7%

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

   11. Taylor expanded in K around 0 41.3%

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

       1. exp-neg 41.3%

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

       2. associate-*r/ 41.3%

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

       3. *-rgt-identity 41.3%

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

       4. cos-neg 41.3%

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

   13. Simplified 41.3%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.8 \cdot 10^{+39} \lor \neg \left(m \leq 7.8 \cdot 10^{+88}\right):\\ \;\;\;\;e^{\left|n - m\right| - 0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \]

Alternative 7: 7.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

   1. +-commutative 75.3%

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

   2. +-commutative 75.3%

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

   3. fabs-sub 75.3%

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

   4. associate-/l* 75.4%

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

   5. +-commutative 75.4%

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

3. Simplified 75.4%

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

4. Taylor expanded in K around 0 95.5%

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

   1. cos-neg 95.5%

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

6. Simplified 95.5%

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

7. Taylor expanded in n around inf 51.5%

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

   1. *-commutative 51.5%

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

   2. unpow2 51.5%

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

9. Simplified 51.5%

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

10. Taylor expanded in n around 0 8.2%

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

11. Taylor expanded in M around 0 8.6%

    \[\leadsto \color{blue}{e^{\left|n - m\right|}} \]

12. Final simplification 8.6%

    \[\leadsto e^{\left|n - m\right|} \]

Alternative 8: 36.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{\cos M}{e^{\ell}} \end{array} \]

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return cos(M) / 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 = cos(m_1) / exp(l)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

   1. +-commutative 75.3%

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

   2. +-commutative 75.3%

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

   3. fabs-sub 75.3%

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

   4. associate-/l* 75.4%

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

   5. +-commutative 75.4%

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

3. Simplified 75.4%

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

4. Taylor expanded in l around inf 21.4%

   \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell} + \left|n - m\right|} \]
5. Step-by-step derivation

   1. neg-mul-1 21.4%

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

6. Simplified 21.4%

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

7. Taylor expanded in l around inf 27.5%

   \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
8. Step-by-step derivation

   1. neg-mul-1 27.5%

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

9. Simplified 27.5%

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

10. Taylor expanded in m around 0 28.6%

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

11. Taylor expanded in K around 0 34.7%

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

    1. exp-neg 34.7%

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

    2. associate-*r/ 34.7%

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

    3. *-rgt-identity 34.7%

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

    4. cos-neg 34.7%

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

13. Simplified 34.7%

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

14. Final simplification 34.7%

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

Alternative 9: 7.1% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

   1. +-commutative 75.3%

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

   2. +-commutative 75.3%

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

   3. fabs-sub 75.3%

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

   4. associate-/l* 75.4%

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

   5. +-commutative 75.4%

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

3. Simplified 75.4%

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

4. Taylor expanded in l around inf 21.4%

   \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell} + \left|n - m\right|} \]
5. Step-by-step derivation

   1. neg-mul-1 21.4%

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

6. Simplified 21.4%

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

7. Taylor expanded in l around inf 27.5%

   \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{-1 \cdot \ell}} \]
8. Step-by-step derivation

   1. neg-mul-1 27.5%

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

9. Simplified 27.5%

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

10. Taylor expanded in l around 0 5.8%

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

11. Taylor expanded in K around 0 6.7%

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

    1. cos-neg 6.7%

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

13. Simplified 6.7%

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

14. Final simplification 6.7%

    \[\leadsto \cos M \]

Reproduce

herbie shell --seed 2023279 
(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)))))))