Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.9% → 96.6%

Time: 15.0s

Alternatives: 11

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 96.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

   1. *-commutative 77.9%

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

   2. associate-*r/ 77.9%

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

   3. associate--r- 77.9%

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

   4. +-commutative 77.9%

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

   5. associate-+r- 77.9%

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

   6. unsub-neg 77.9%

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

   7. associate--r+ 77.9%

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

   8. +-commutative 77.9%

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

   9. associate--r+ 77.9%

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

3. Simplified 77.9%

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

4. Taylor expanded in K around 0 94.3%

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

   1. cos-neg 94.3%

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

6. Simplified 94.3%

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

7. Final simplification 94.3%

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

Alternative 2: 68.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|m - n\right|\\ t_1 := t_0 - \ell\\ \mathbf{if}\;n \leq 4.2 \cdot 10^{-257}:\\ \;\;\;\;\cos M \cdot e^{t_1 + -0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{t_0 - \left(\ell + M \cdot M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t_1 - \left(n \cdot n\right) \cdot 0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- m n))) (t_1 (- t_0 l)))
   (if (<= n 4.2e-257)
     (* (cos M) (exp (+ t_1 (* -0.25 (* m m)))))
     (if (<= n 5.5e-6)
       (* (cos (- (/ K (/ 2.0 (+ m n))) M)) (exp (- t_0 (+ l (* M M)))))
       (* (cos M) (exp (- t_1 (* (* n n) 0.25))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((m - n));
	double t_1 = t_0 - l;
	double tmp;
	if (n <= 4.2e-257) {
		tmp = cos(M) * exp((t_1 + (-0.25 * (m * m))));
	} else if (n <= 5.5e-6) {
		tmp = cos(((K / (2.0 / (m + n))) - M)) * exp((t_0 - (l + (M * M))));
	} else {
		tmp = cos(M) * exp((t_1 - ((n * n) * 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) :: t_1
    real(8) :: tmp
    t_0 = abs((m - n))
    t_1 = t_0 - l
    if (n <= 4.2d-257) then
        tmp = cos(m_1) * exp((t_1 + ((-0.25d0) * (m * m))))
    else if (n <= 5.5d-6) then
        tmp = cos(((k / (2.0d0 / (m + n))) - m_1)) * exp((t_0 - (l + (m_1 * m_1))))
    else
        tmp = cos(m_1) * exp((t_1 - ((n * n) * 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((m - n));
	double t_1 = t_0 - l;
	double tmp;
	if (n <= 4.2e-257) {
		tmp = Math.cos(M) * Math.exp((t_1 + (-0.25 * (m * m))));
	} else if (n <= 5.5e-6) {
		tmp = Math.cos(((K / (2.0 / (m + n))) - M)) * Math.exp((t_0 - (l + (M * M))));
	} else {
		tmp = Math.cos(M) * Math.exp((t_1 - ((n * n) * 0.25)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((m - n))
	t_1 = t_0 - l
	tmp = 0
	if n <= 4.2e-257:
		tmp = math.cos(M) * math.exp((t_1 + (-0.25 * (m * m))))
	elif n <= 5.5e-6:
		tmp = math.cos(((K / (2.0 / (m + n))) - M)) * math.exp((t_0 - (l + (M * M))))
	else:
		tmp = math.cos(M) * math.exp((t_1 - ((n * n) * 0.25)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(m - n))
	t_1 = Float64(t_0 - l)
	tmp = 0.0
	if (n <= 4.2e-257)
		tmp = Float64(cos(M) * exp(Float64(t_1 + Float64(-0.25 * Float64(m * m)))));
	elseif (n <= 5.5e-6)
		tmp = Float64(cos(Float64(Float64(K / Float64(2.0 / Float64(m + n))) - M)) * exp(Float64(t_0 - Float64(l + Float64(M * M)))));
	else
		tmp = Float64(cos(M) * exp(Float64(t_1 - Float64(Float64(n * n) * 0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((m - n));
	t_1 = t_0 - l;
	tmp = 0.0;
	if (n <= 4.2e-257)
		tmp = cos(M) * exp((t_1 + (-0.25 * (m * m))));
	elseif (n <= 5.5e-6)
		tmp = cos(((K / (2.0 / (m + n))) - M)) * exp((t_0 - (l + (M * M))));
	else
		tmp = cos(M) * exp((t_1 - ((n * n) * 0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - l), $MachinePrecision]}, If[LessEqual[n, 4.2e-257], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$1 + N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.5e-6], N[(N[Cos[N[(N[(K / N[(2.0 / N[(m + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(l + N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$1 - N[(N[(n * n), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if n < 4.2000000000000002e-257

   1. Initial program 78.4%

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

      1. *-commutative 78.4%

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

      2. associate-*r/ 78.4%

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

      3. associate--r- 78.4%

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

      4. +-commutative 78.4%

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

      5. associate-+r- 78.4%

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

      6. unsub-neg 78.4%

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

      7. associate--r+ 78.4%

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

      8. +-commutative 78.4%

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

      9. associate--r+ 78.4%

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

   3. Simplified 78.4%

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

   4. Taylor expanded in m around inf 50.5%

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

      1. *-commutative 50.5%

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

      2. unpow2 50.5%

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

   6. Simplified 50.5%

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

   7. Taylor expanded in K around 0 61.2%

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

      1. *-commutative 61.2%

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

   9. Simplified 61.2%

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

   if 4.2000000000000002e-257 < n < 5.4999999999999999e-6

   1. Initial program 77.2%

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

      1. associate-/l* 77.5%

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

      2. associate--r- 77.5%

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

   3. Simplified 77.5%

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

   4. Taylor expanded in M around inf 62.4%

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

      1. unpow2 66.1%

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

   6. Simplified 62.4%

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

   if 5.4999999999999999e-6 < n

   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%

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

      2. associate-*r/ 77.6%

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

      3. associate--r- 77.6%

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

      4. +-commutative 77.6%

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

      5. associate-+r- 77.6%

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

      6. unsub-neg 77.6%

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

      7. associate--r+ 77.6%

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

      8. +-commutative 77.6%

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

      9. associate--r+ 77.6%

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

   3. Simplified 77.6%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around inf 87.9%

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

      1. *-commutative 87.9%

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

      2. unpow2 87.9%

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

   9. Simplified 87.9%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 4.2 \cdot 10^{-257}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) + -0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left|m - n\right| - \left(\ell + M \cdot M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(n \cdot n\right) \cdot 0.25}\\ \end{array} \]

Alternative 3: 72.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|m - n\right| - \ell\\ \mathbf{if}\;m \leq -1000000:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq 1.85 \cdot 10^{-239}:\\ \;\;\;\;\cos M \cdot e^{t_0 - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t_0 - \left(n \cdot n\right) \cdot 0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (fabs (- m n)) l)))
   (if (<= m -1000000.0)
     (* (cos M) (exp (* -0.25 (* m m))))
     (if (<= m 1.85e-239)
       (* (cos M) (exp (- t_0 (* M M))))
       (* (cos M) (exp (- t_0 (* (* n n) 0.25))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((m - n)) - l;
	double tmp;
	if (m <= -1000000.0) {
		tmp = cos(M) * exp((-0.25 * (m * m)));
	} else if (m <= 1.85e-239) {
		tmp = cos(M) * exp((t_0 - (M * M)));
	} else {
		tmp = cos(M) * exp((t_0 - ((n * n) * 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((m - n)) - l
    if (m <= (-1000000.0d0)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m * m)))
    else if (m <= 1.85d-239) then
        tmp = cos(m_1) * exp((t_0 - (m_1 * m_1)))
    else
        tmp = cos(m_1) * exp((t_0 - ((n * n) * 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((m - n)) - l;
	double tmp;
	if (m <= -1000000.0) {
		tmp = Math.cos(M) * Math.exp((-0.25 * (m * m)));
	} else if (m <= 1.85e-239) {
		tmp = Math.cos(M) * Math.exp((t_0 - (M * M)));
	} else {
		tmp = Math.cos(M) * Math.exp((t_0 - ((n * n) * 0.25)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((m - n)) - l
	tmp = 0
	if m <= -1000000.0:
		tmp = math.cos(M) * math.exp((-0.25 * (m * m)))
	elif m <= 1.85e-239:
		tmp = math.cos(M) * math.exp((t_0 - (M * M)))
	else:
		tmp = math.cos(M) * math.exp((t_0 - ((n * n) * 0.25)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(abs(Float64(m - n)) - l)
	tmp = 0.0
	if (m <= -1000000.0)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(m * m))));
	elseif (m <= 1.85e-239)
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(M * M))));
	else
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(Float64(n * n) * 0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((m - n)) - l;
	tmp = 0.0;
	if (m <= -1000000.0)
		tmp = cos(M) * exp((-0.25 * (m * m)));
	elseif (m <= 1.85e-239)
		tmp = cos(M) * exp((t_0 - (M * M)));
	else
		tmp = cos(M) * exp((t_0 - ((n * n) * 0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]}, If[LessEqual[m, -1000000.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.85e-239], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(N[(n * n), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < -1e6

   1. Initial program 72.4%

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

      1. *-commutative 72.4%

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

      2. associate-*r/ 72.4%

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

      3. associate--r- 72.4%

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

      4. +-commutative 72.4%

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

      5. associate-+r- 72.4%

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

      6. unsub-neg 72.4%

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

      7. associate--r+ 72.4%

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

      8. +-commutative 72.4%

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

      9. associate--r+ 72.4%

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

   3. Simplified 72.4%

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

   4. Taylor expanded in K around 0 98.7%

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

      1. cos-neg 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in m around inf 97.4%

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

      1. *-commutative 97.4%

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

      2. unpow2 97.4%

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

   9. Simplified 97.4%

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

   if -1e6 < m < 1.85000000000000008e-239

   1. Initial program 77.2%

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

      1. *-commutative 77.2%

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

      2. associate-*r/ 77.2%

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

      3. associate--r- 77.2%

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

      4. +-commutative 77.2%

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

      5. associate-+r- 77.2%

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

      6. unsub-neg 77.2%

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

      7. associate--r+ 77.2%

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

      8. +-commutative 77.2%

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

      9. associate--r+ 77.2%

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

   3. Simplified 77.2%

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

   4. Taylor expanded in K around 0 90.2%

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

      1. cos-neg 90.2%

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

   6. Simplified 90.2%

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

   7. Taylor expanded in M around inf 63.2%

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

      1. unpow2 63.2%

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

   9. Simplified 63.2%

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

   if 1.85000000000000008e-239 < m

   1. Initial program 82.4%

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

      1. *-commutative 82.4%

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

      2. associate-*r/ 82.4%

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

      3. associate--r- 82.4%

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

      4. +-commutative 82.4%

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

      5. associate-+r- 82.4%

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

      6. unsub-neg 82.4%

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

      7. associate--r+ 82.4%

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

      8. +-commutative 82.4%

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

      9. associate--r+ 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in K around 0 94.1%

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

      1. cos-neg 94.1%

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

   6. Simplified 94.1%

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

   7. Taylor expanded in n around inf 54.9%

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

      1. *-commutative 54.9%

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

      2. unpow2 54.9%

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

   9. Simplified 54.9%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1000000:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq 1.85 \cdot 10^{-239}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(n \cdot n\right) \cdot 0.25}\\ \end{array} \]

Alternative 4: 69.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|m - n\right| - \ell\\ \mathbf{if}\;n \leq 2.8 \cdot 10^{-265}:\\ \;\;\;\;\cos M \cdot e^{t_0 + -0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 21000000:\\ \;\;\;\;\cos M \cdot e^{t_0 - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t_0 - \left(n \cdot n\right) \cdot 0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (fabs (- m n)) l)))
   (if (<= n 2.8e-265)
     (* (cos M) (exp (+ t_0 (* -0.25 (* m m)))))
     (if (<= n 21000000.0)
       (* (cos M) (exp (- t_0 (* M M))))
       (* (cos M) (exp (- t_0 (* (* n n) 0.25))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((m - n)) - l;
	double tmp;
	if (n <= 2.8e-265) {
		tmp = cos(M) * exp((t_0 + (-0.25 * (m * m))));
	} else if (n <= 21000000.0) {
		tmp = cos(M) * exp((t_0 - (M * M)));
	} else {
		tmp = cos(M) * exp((t_0 - ((n * n) * 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((m - n)) - l
    if (n <= 2.8d-265) then
        tmp = cos(m_1) * exp((t_0 + ((-0.25d0) * (m * m))))
    else if (n <= 21000000.0d0) then
        tmp = cos(m_1) * exp((t_0 - (m_1 * m_1)))
    else
        tmp = cos(m_1) * exp((t_0 - ((n * n) * 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((m - n)) - l;
	double tmp;
	if (n <= 2.8e-265) {
		tmp = Math.cos(M) * Math.exp((t_0 + (-0.25 * (m * m))));
	} else if (n <= 21000000.0) {
		tmp = Math.cos(M) * Math.exp((t_0 - (M * M)));
	} else {
		tmp = Math.cos(M) * Math.exp((t_0 - ((n * n) * 0.25)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((m - n)) - l
	tmp = 0
	if n <= 2.8e-265:
		tmp = math.cos(M) * math.exp((t_0 + (-0.25 * (m * m))))
	elif n <= 21000000.0:
		tmp = math.cos(M) * math.exp((t_0 - (M * M)))
	else:
		tmp = math.cos(M) * math.exp((t_0 - ((n * n) * 0.25)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(abs(Float64(m - n)) - l)
	tmp = 0.0
	if (n <= 2.8e-265)
		tmp = Float64(cos(M) * exp(Float64(t_0 + Float64(-0.25 * Float64(m * m)))));
	elseif (n <= 21000000.0)
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(M * M))));
	else
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(Float64(n * n) * 0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((m - n)) - l;
	tmp = 0.0;
	if (n <= 2.8e-265)
		tmp = cos(M) * exp((t_0 + (-0.25 * (m * m))));
	elseif (n <= 21000000.0)
		tmp = cos(M) * exp((t_0 - (M * M)));
	else
		tmp = cos(M) * exp((t_0 - ((n * n) * 0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]}, If[LessEqual[n, 2.8e-265], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 + N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 21000000.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(N[(n * n), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < 2.80000000000000023e-265

   1. Initial program 78.9%

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

      1. *-commutative 78.9%

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

      2. associate-*r/ 78.9%

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

      3. associate--r- 78.9%

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

      4. +-commutative 78.9%

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

      5. associate-+r- 78.9%

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

      6. unsub-neg 78.9%

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

      7. associate--r+ 78.9%

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

      8. +-commutative 78.9%

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

      9. associate--r+ 78.9%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in m around inf 50.2%

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

      1. *-commutative 50.2%

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

      2. unpow2 50.2%

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

   6. Simplified 50.2%

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

   7. Taylor expanded in K around 0 61.3%

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

      1. *-commutative 61.3%

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

   9. Simplified 61.3%

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

   if 2.80000000000000023e-265 < n < 2.1e7

   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%

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

      2. associate-*r/ 75.4%

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

      3. associate--r- 75.4%

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

      4. +-commutative 75.4%

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

      5. associate-+r- 75.4%

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

      6. unsub-neg 75.4%

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

      7. associate--r+ 75.4%

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

      8. +-commutative 75.4%

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

      9. associate--r+ 75.4%

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

   3. Simplified 75.4%

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

   4. Taylor expanded in K around 0 87.6%

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

      1. cos-neg 87.6%

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

   6. Simplified 87.6%

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

   7. Taylor expanded in M around inf 63.7%

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

      1. unpow2 63.7%

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

   9. Simplified 63.7%

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

   if 2.1e7 < n

   1. Initial program 78.7%

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

      1. *-commutative 78.7%

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

      2. associate-*r/ 78.7%

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

      3. associate--r- 78.7%

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

      4. +-commutative 78.7%

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

      5. associate-+r- 78.7%

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

      6. unsub-neg 78.7%

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

      7. associate--r+ 78.7%

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

      8. +-commutative 78.7%

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

      9. associate--r+ 78.7%

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

   3. Simplified 78.7%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around inf 91.6%

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

      1. *-commutative 91.6%

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

      2. unpow2 91.6%

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

   9. Simplified 91.6%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2.8 \cdot 10^{-265}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) + -0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 21000000:\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(n \cdot n\right) \cdot 0.25}\\ \end{array} \]

Alternative 5: 72.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|m - n\right| - \ell\\ \mathbf{if}\;m \leq -6800:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq 3.5 \cdot 10^{-239}:\\ \;\;\;\;\cos M \cdot e^{t_0 - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{t_0 + -0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (fabs (- m n)) l)))
   (if (<= m -6800.0)
     (* (cos M) (exp (* -0.25 (* m m))))
     (if (<= m 3.5e-239)
       (* (cos M) (exp (- t_0 (* M M))))
       (exp (+ t_0 (* -0.25 (* n n))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((m - n)) - l;
	double tmp;
	if (m <= -6800.0) {
		tmp = cos(M) * exp((-0.25 * (m * m)));
	} else if (m <= 3.5e-239) {
		tmp = cos(M) * exp((t_0 - (M * M)));
	} else {
		tmp = exp((t_0 + (-0.25 * (n * n))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((m - n)) - l
    if (m <= (-6800.0d0)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m * m)))
    else if (m <= 3.5d-239) then
        tmp = cos(m_1) * exp((t_0 - (m_1 * m_1)))
    else
        tmp = exp((t_0 + ((-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.abs((m - n)) - l;
	double tmp;
	if (m <= -6800.0) {
		tmp = Math.cos(M) * Math.exp((-0.25 * (m * m)));
	} else if (m <= 3.5e-239) {
		tmp = Math.cos(M) * Math.exp((t_0 - (M * M)));
	} else {
		tmp = Math.exp((t_0 + (-0.25 * (n * n))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((m - n)) - l
	tmp = 0
	if m <= -6800.0:
		tmp = math.cos(M) * math.exp((-0.25 * (m * m)))
	elif m <= 3.5e-239:
		tmp = math.cos(M) * math.exp((t_0 - (M * M)))
	else:
		tmp = math.exp((t_0 + (-0.25 * (n * n))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(abs(Float64(m - n)) - l)
	tmp = 0.0
	if (m <= -6800.0)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(m * m))));
	elseif (m <= 3.5e-239)
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(M * M))));
	else
		tmp = exp(Float64(t_0 + Float64(-0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((m - n)) - l;
	tmp = 0.0;
	if (m <= -6800.0)
		tmp = cos(M) * exp((-0.25 * (m * m)));
	elseif (m <= 3.5e-239)
		tmp = cos(M) * exp((t_0 - (M * M)));
	else
		tmp = exp((t_0 + (-0.25 * (n * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]}, If[LessEqual[m, -6800.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 3.5e-239], 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[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < -6800

   1. Initial program 72.4%

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

      1. *-commutative 72.4%

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

      2. associate-*r/ 72.4%

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

      3. associate--r- 72.4%

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

      4. +-commutative 72.4%

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

      5. associate-+r- 72.4%

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

      6. unsub-neg 72.4%

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

      7. associate--r+ 72.4%

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

      8. +-commutative 72.4%

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

      9. associate--r+ 72.4%

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

   3. Simplified 72.4%

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

   4. Taylor expanded in K around 0 98.7%

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

      1. cos-neg 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in m around inf 97.4%

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

      1. *-commutative 97.4%

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

      2. unpow2 97.4%

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

   9. Simplified 97.4%

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

   if -6800 < m < 3.50000000000000005e-239

   1. Initial program 77.2%

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

      1. *-commutative 77.2%

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

      2. associate-*r/ 77.2%

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

      3. associate--r- 77.2%

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

      4. +-commutative 77.2%

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

      5. associate-+r- 77.2%

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

      6. unsub-neg 77.2%

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

      7. associate--r+ 77.2%

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

      8. +-commutative 77.2%

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

      9. associate--r+ 77.2%

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

   3. Simplified 77.2%

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

   4. Taylor expanded in K around 0 90.2%

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

      1. cos-neg 90.2%

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

   6. Simplified 90.2%

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

   7. Taylor expanded in M around inf 63.2%

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

      1. unpow2 63.2%

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

   9. Simplified 63.2%

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

   if 3.50000000000000005e-239 < m

   1. Initial program 82.4%

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

      1. *-commutative 82.4%

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

      2. associate-*r/ 82.4%

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

      3. associate--r- 82.4%

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

      4. +-commutative 82.4%

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

      5. associate-+r- 82.4%

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

      6. unsub-neg 82.4%

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

      7. associate--r+ 82.4%

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

      8. +-commutative 82.4%

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

      9. associate--r+ 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in K around 0 94.1%

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

      1. cos-neg 94.1%

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

   6. Simplified 94.1%

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

   7. Taylor expanded in n around inf 54.9%

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

      1. *-commutative 54.9%

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

      2. unpow2 54.9%

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

   9. Simplified 54.9%

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

   10. Taylor expanded in M around 0 54.0%

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

       1. exp-diff 17.4%

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

       2. sub-neg 17.4%

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

       3. mul-1-neg 17.4%

          \[\leadsto \frac{e^{\left|m + \color{blue}{-1 \cdot n}\right|}}{e^{\ell + 0.25 \cdot {n}^{2}}} \]

       4. +-commutative 17.4%

          \[\leadsto \frac{e^{\left|\color{blue}{-1 \cdot n + m}\right|}}{e^{\ell + 0.25 \cdot {n}^{2}}} \]

       5. remove-double-neg 17.4%

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

       6. mul-1-neg 17.4%

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

       7. mul-1-neg 17.4%

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

       8. distribute-neg-in 17.4%

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

       9. exp-diff 54.0%

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

       10. associate--r+ 54.0%

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

       11. *-commutative 54.0%

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

       12. unpow2 54.0%

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

       13. associate-*r* 54.0%

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

   12. Simplified 54.0%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6800:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq 3.5 \cdot 10^{-239}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) + -0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]

Alternative 6: 70.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 9.5 \cdot 10^{-229}:\\ \;\;\;\;e^{\mathsf{fma}\left(-0.25, m \cdot m, \left|m - n\right|\right) - \ell}\\ \mathbf{elif}\;n \leq 55:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 9.5e-229)
   (exp (- (fma -0.25 (* m m) (fabs (- m n))) l))
   (if (<= n 55.0)
     (* (cos M) (exp (* M (- M))))
     (* (cos M) (exp (* -0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 9.5e-229) {
		tmp = exp((fma(-0.25, (m * m), fabs((m - n))) - l));
	} else if (n <= 55.0) {
		tmp = cos(M) * exp((M * -M));
	} else {
		tmp = cos(M) * exp((-0.25 * (n * n)));
	}
	return tmp;
}

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 9.5e-229)
		tmp = exp(Float64(fma(-0.25, Float64(m * m), abs(Float64(m - n))) - l));
	elseif (n <= 55.0)
		tmp = Float64(cos(M) * exp(Float64(M * Float64(-M))));
	else
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(n * n))));
	end
	return tmp
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 9.5e-229], N[Exp[N[(N[(-0.25 * N[(m * m), $MachinePrecision] + N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 55.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 9.4999999999999997e-229

   1. Initial program 79.8%

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

      1. *-commutative 79.8%

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

      2. associate-*r/ 79.8%

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

      3. associate--r- 79.8%

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

      4. +-commutative 79.8%

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

      5. associate-+r- 79.8%

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

      6. unsub-neg 79.8%

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

      7. associate--r+ 79.8%

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

      8. +-commutative 79.8%

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

      9. associate--r+ 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in m around inf 53.0%

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

      1. *-commutative 53.0%

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

      2. unpow2 53.0%

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

   6. Simplified 53.0%

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

   7. Taylor expanded in K around 0 63.1%

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

      1. *-commutative 63.1%

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

   9. Simplified 63.1%

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

   10. Taylor expanded in M around 0 61.8%

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

       1. exp-diff 50.8%

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

       2. rem-log-exp 50.8%

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

       3. rem-log-exp 50.8%

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

       4. exp-diff 50.8%

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

       5. rem-log-exp 58.5%

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

       6. fma-def 58.5%

          \[\leadsto e^{\color{blue}{\mathsf{fma}\left(-0.25, {m}^{2}, \left|m - n\right|\right)} - \log \left(e^{\ell}\right)} \]

       7. unpow2 58.5%

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

       8. rem-log-exp 61.8%

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

   12. Simplified 61.8%

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

   if 9.4999999999999997e-229 < n < 55

   1. Initial program 73.2%

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

      1. *-commutative 73.2%

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

      2. associate-*r/ 73.2%

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

      3. associate--r- 73.2%

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

      4. +-commutative 73.2%

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

      5. associate-+r- 73.2%

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

      6. unsub-neg 73.2%

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

      7. associate--r+ 73.2%

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

      8. +-commutative 73.2%

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

      9. associate--r+ 73.2%

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

   3. Simplified 73.2%

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

   4. Taylor expanded in K around 0 87.4%

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

      1. cos-neg 87.4%

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

   6. Simplified 87.4%

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

   7. Taylor expanded in M around inf 52.7%

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

      1. mul-1-neg 52.7%

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

      2. unpow2 52.7%

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

      3. distribute-rgt-neg-in 52.7%

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

   9. Simplified 52.7%

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

   if 55 < n

   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%

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

      2. associate-*r/ 77.1%

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

      3. associate--r- 77.1%

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

      4. +-commutative 77.1%

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

      5. associate-+r- 77.1%

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

      6. unsub-neg 77.1%

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

      7. associate--r+ 77.1%

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

      8. +-commutative 77.1%

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

      9. associate--r+ 77.1%

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

   3. Simplified 77.1%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around inf 95.9%

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

      1. *-commutative 95.9%

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

      2. unpow2 95.9%

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

   9. Simplified 95.9%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 9.5 \cdot 10^{-229}:\\ \;\;\;\;e^{\mathsf{fma}\left(-0.25, m \cdot m, \left|m - n\right|\right) - \ell}\\ \mathbf{elif}\;n \leq 55:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]

Alternative 7: 73.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.55 \cdot 10^{+25}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) + -0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -1.55e+25)
   (* (cos M) (exp (* -0.25 (* m m))))
   (exp (+ (- (fabs (- m n)) l) (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1.55e+25) {
		tmp = cos(M) * exp((-0.25 * (m * m)));
	} else {
		tmp = exp(((fabs((m - n)) - l) + (-0.25 * (n * n))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-1.55d+25)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m * m)))
    else
        tmp = exp(((abs((m - n)) - l) + ((-0.25d0) * (n * n))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1.55e+25) {
		tmp = Math.cos(M) * Math.exp((-0.25 * (m * m)));
	} else {
		tmp = Math.exp(((Math.abs((m - n)) - l) + (-0.25 * (n * n))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -1.55e+25:
		tmp = math.cos(M) * math.exp((-0.25 * (m * m)))
	else:
		tmp = math.exp(((math.fabs((m - n)) - l) + (-0.25 * (n * n))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -1.55e+25)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(m * m))));
	else
		tmp = exp(Float64(Float64(abs(Float64(m - n)) - l) + Float64(-0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -1.55e+25)
		tmp = cos(M) * exp((-0.25 * (m * m)));
	else
		tmp = exp(((abs((m - n)) - l) + (-0.25 * (n * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -1.55e+25], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] + N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.5499999999999999e25

   1. Initial program 71.0%

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

      1. *-commutative 71.0%

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

      2. associate-*r/ 71.0%

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

      3. associate--r- 71.0%

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

      4. +-commutative 71.0%

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

      5. associate-+r- 71.0%

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

      6. unsub-neg 71.0%

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

      7. associate--r+ 71.0%

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

      8. +-commutative 71.0%

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

      9. associate--r+ 71.0%

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

   3. Simplified 71.0%

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

   4. Taylor expanded in K around 0 98.6%

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

      1. cos-neg 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in m around inf 98.6%

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

      1. *-commutative 98.6%

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

      2. unpow2 98.6%

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

   9. Simplified 98.6%

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

   if -1.5499999999999999e25 < m

   1. Initial program 80.5%

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

      1. *-commutative 80.5%

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

      2. associate-*r/ 80.5%

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

      3. associate--r- 80.5%

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

      4. +-commutative 80.5%

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

      5. associate-+r- 80.5%

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

      6. unsub-neg 80.5%

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

      7. associate--r+ 80.5%

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

      8. +-commutative 80.5%

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

      9. associate--r+ 80.5%

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

   3. Simplified 80.5%

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

   4. Taylor expanded in K around 0 92.8%

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

      1. cos-neg 92.8%

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

   6. Simplified 92.8%

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

   7. Taylor expanded in n around inf 61.7%

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

      1. *-commutative 61.7%

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

      2. unpow2 61.7%

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

   9. Simplified 61.7%

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

   10. Taylor expanded in M around 0 60.0%

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

       1. exp-diff 18.2%

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

       2. sub-neg 18.2%

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

       3. mul-1-neg 18.2%

          \[\leadsto \frac{e^{\left|m + \color{blue}{-1 \cdot n}\right|}}{e^{\ell + 0.25 \cdot {n}^{2}}} \]

       4. +-commutative 18.2%

          \[\leadsto \frac{e^{\left|\color{blue}{-1 \cdot n + m}\right|}}{e^{\ell + 0.25 \cdot {n}^{2}}} \]

       5. remove-double-neg 18.2%

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

       6. mul-1-neg 18.2%

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

       7. mul-1-neg 18.2%

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

       8. distribute-neg-in 18.2%

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

       9. exp-diff 60.0%

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

       10. associate--r+ 60.0%

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

       11. *-commutative 60.0%

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

       12. unpow2 60.0%

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

       13. associate-*r* 60.0%

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

   12. Simplified 60.0%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.55 \cdot 10^{+25}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) + -0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]

Alternative 8: 76.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -2.25 \cdot 10^{-5} \lor \neg \left(M \leq 1.1 \cdot 10^{-5}\right):\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -2.25e-5) (not (<= M 1.1e-5)))
   (* (cos M) (exp (* M (- M))))
   (* (cos M) (exp (* -0.25 (* m m))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -2.25e-5) || !(M <= 1.1e-5)) {
		tmp = cos(M) * exp((M * -M));
	} else {
		tmp = cos(M) * exp((-0.25 * (m * m)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((m_1 <= (-2.25d-5)) .or. (.not. (m_1 <= 1.1d-5))) then
        tmp = cos(m_1) * exp((m_1 * -m_1))
    else
        tmp = cos(m_1) * exp(((-0.25d0) * (m * m)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -2.25e-5) || !(M <= 1.1e-5)) {
		tmp = Math.cos(M) * Math.exp((M * -M));
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * (m * m)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -2.25e-5) or not (M <= 1.1e-5):
		tmp = math.cos(M) * math.exp((M * -M))
	else:
		tmp = math.cos(M) * math.exp((-0.25 * (m * m)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -2.25e-5) || !(M <= 1.1e-5))
		tmp = Float64(cos(M) * exp(Float64(M * Float64(-M))));
	else
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(m * m))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -2.25e-5) || ~((M <= 1.1e-5)))
		tmp = cos(M) * exp((M * -M));
	else
		tmp = cos(M) * exp((-0.25 * (m * m)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -2.25e-5], N[Not[LessEqual[M, 1.1e-5]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -2.25000000000000014e-5 or 1.1e-5 < M

   1. Initial program 82.0%

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

      1. *-commutative 82.0%

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

      2. associate-*r/ 82.0%

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

      3. associate--r- 82.0%

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

      4. +-commutative 82.0%

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

      5. associate-+r- 82.0%

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

      6. unsub-neg 82.0%

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

      7. associate--r+ 82.0%

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

      8. +-commutative 82.0%

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

      9. associate--r+ 82.0%

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

   3. Simplified 82.0%

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

   4. Taylor expanded in K around 0 99.4%

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

      1. cos-neg 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in M around inf 95.4%

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

      1. mul-1-neg 95.4%

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

      2. unpow2 95.4%

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

      3. distribute-rgt-neg-in 95.4%

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

   9. Simplified 95.4%

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

   if -2.25000000000000014e-5 < M < 1.1e-5

   1. Initial program 73.8%

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

      1. *-commutative 73.8%

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

      2. associate-*r/ 73.8%

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

      3. associate--r- 73.8%

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

      4. +-commutative 73.8%

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

      5. associate-+r- 73.8%

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

      6. unsub-neg 73.8%

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

      7. associate--r+ 73.8%

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

      8. +-commutative 73.8%

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

      9. associate--r+ 73.8%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in K around 0 89.3%

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

      1. cos-neg 89.3%

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

   6. Simplified 89.3%

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

   7. Taylor expanded in m around inf 56.5%

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

      1. *-commutative 56.5%

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

      2. unpow2 56.5%

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

   9. Simplified 56.5%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2.25 \cdot 10^{-5} \lor \neg \left(M \leq 1.1 \cdot 10^{-5}\right):\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \end{array} \]

Alternative 9: 65.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 1.05 \cdot 10^{-266}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 55:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 1.05e-266)
   (* (cos M) (exp (* -0.25 (* m m))))
   (if (<= n 55.0)
     (* (cos M) (exp (* M (- M))))
     (* (cos M) (exp (* -0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1.05e-266) {
		tmp = cos(M) * exp((-0.25 * (m * m)));
	} else if (n <= 55.0) {
		tmp = cos(M) * exp((M * -M));
	} else {
		tmp = cos(M) * exp((-0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 1.05d-266) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m * m)))
    else if (n <= 55.0d0) then
        tmp = cos(m_1) * exp((m_1 * -m_1))
    else
        tmp = cos(m_1) * exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1.05e-266) {
		tmp = Math.cos(M) * Math.exp((-0.25 * (m * m)));
	} else if (n <= 55.0) {
		tmp = Math.cos(M) * Math.exp((M * -M));
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 1.05e-266:
		tmp = math.cos(M) * math.exp((-0.25 * (m * m)))
	elif n <= 55.0:
		tmp = math.cos(M) * math.exp((M * -M))
	else:
		tmp = math.cos(M) * math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 1.05e-266)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(m * m))));
	elseif (n <= 55.0)
		tmp = Float64(cos(M) * exp(Float64(M * Float64(-M))));
	else
		tmp = Float64(cos(M) * exp(Float64(-0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 1.05e-266)
		tmp = cos(M) * exp((-0.25 * (m * m)));
	elseif (n <= 55.0)
		tmp = cos(M) * exp((M * -M));
	else
		tmp = cos(M) * exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 1.05e-266], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 55.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 1.04999999999999998e-266

   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%

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

      2. associate-*r/ 78.8%

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

      3. associate--r- 78.8%

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

      4. +-commutative 78.8%

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

      5. associate-+r- 78.8%

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

      6. unsub-neg 78.8%

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

      7. associate--r+ 78.8%

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

      8. +-commutative 78.8%

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

      9. associate--r+ 78.8%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in K around 0 95.6%

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

      1. cos-neg 95.6%

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

   6. Simplified 95.6%

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

   7. Taylor expanded in m around inf 59.9%

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

      1. *-commutative 59.9%

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

      2. unpow2 59.9%

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

   9. Simplified 59.9%

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

   if 1.04999999999999998e-266 < n < 55

   1. Initial program 76.8%

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

      1. *-commutative 76.8%

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

      2. associate-*r/ 76.8%

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

      3. associate--r- 76.8%

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

      4. +-commutative 76.8%

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

      5. associate-+r- 76.8%

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

      6. unsub-neg 76.8%

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

      7. associate--r+ 76.8%

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

      8. +-commutative 76.8%

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

      9. associate--r+ 76.8%

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

   3. Simplified 76.8%

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

   4. Taylor expanded in K around 0 87.6%

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

      1. cos-neg 87.6%

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

   6. Simplified 87.6%

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

   7. Taylor expanded in M around inf 53.5%

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

      1. mul-1-neg 53.5%

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

      2. unpow2 53.5%

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

      3. distribute-rgt-neg-in 53.5%

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

   9. Simplified 53.5%

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

   if 55 < n

   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%

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

      2. associate-*r/ 77.1%

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

      3. associate--r- 77.1%

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

      4. +-commutative 77.1%

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

      5. associate-+r- 77.1%

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

      6. unsub-neg 77.1%

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

      7. associate--r+ 77.1%

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

      8. +-commutative 77.1%

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

      9. associate--r+ 77.1%

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

   3. Simplified 77.1%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around inf 95.9%

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

      1. *-commutative 95.9%

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

      2. unpow2 95.9%

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

   9. Simplified 95.9%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.05 \cdot 10^{-266}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 55:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]

Alternative 10: 66.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -1.05 \cdot 10^{-6} \lor \neg \left(M \leq 1.3 \cdot 10^{+81}\right):\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -1.05e-6) (not (<= M 1.3e+81)))
   (* (cos M) (exp (* M (- M))))
   (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -1.05e-6) || !(M <= 1.3e+81)) {
		tmp = cos(M) * exp((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 <= (-1.05d-6)) .or. (.not. (m_1 <= 1.3d+81))) then
        tmp = cos(m_1) * exp((m_1 * -m_1))
    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 <= -1.05e-6) || !(M <= 1.3e+81)) {
		tmp = Math.cos(M) * Math.exp((M * -M));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -1.05e-6) or not (M <= 1.3e+81):
		tmp = math.cos(M) * math.exp((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 <= -1.05e-6) || !(M <= 1.3e+81))
		tmp = Float64(cos(M) * exp(Float64(M * Float64(-M))));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -1.05e-6) || ~((M <= 1.3e+81)))
		tmp = cos(M) * exp((M * -M));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -1.05e-6], N[Not[LessEqual[M, 1.3e+81]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -1.0499999999999999e-6 or 1.29999999999999996e81 < M

   1. Initial program 82.1%

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

      1. *-commutative 82.1%

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

      2. associate-*r/ 82.1%

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

      3. associate--r- 82.1%

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

      4. +-commutative 82.1%

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

      5. associate-+r- 82.1%

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

      6. unsub-neg 82.1%

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

      7. associate--r+ 82.1%

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

      8. +-commutative 82.1%

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

      9. associate--r+ 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in M around inf 98.2%

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

      1. mul-1-neg 98.2%

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

      2. unpow2 98.2%

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

      3. distribute-rgt-neg-in 98.2%

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

   9. Simplified 98.2%

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

   if -1.0499999999999999e-6 < M < 1.29999999999999996e81

   1. Initial program 74.5%

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

      1. *-commutative 74.5%

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

      2. associate-*r/ 74.5%

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

      3. associate--r- 74.5%

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

      4. +-commutative 74.5%

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

      5. associate-+r- 74.5%

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

      6. unsub-neg 74.5%

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

      7. associate--r+ 74.5%

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

      8. +-commutative 74.5%

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

      9. associate--r+ 74.5%

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

   3. Simplified 74.5%

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

   4. Taylor expanded in K around 0 89.5%

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

      1. cos-neg 89.5%

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

   6. Simplified 89.5%

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

   7. Taylor expanded in l around inf 39.1%

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

      1. neg-mul-1 39.1%

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

   9. Simplified 39.1%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.05 \cdot 10^{-6} \lor \neg \left(M \leq 1.3 \cdot 10^{+81}\right):\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]

Alternative 11: 36.8% accurate, 2.1× speedup

Math

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

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(Float64(-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 77.9%

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

   1. *-commutative 77.9%

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

   2. associate-*r/ 77.9%

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

   3. associate--r- 77.9%

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

   4. +-commutative 77.9%

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

   5. associate-+r- 77.9%

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

   6. unsub-neg 77.9%

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

   7. associate--r+ 77.9%

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

   8. +-commutative 77.9%

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

   9. associate--r+ 77.9%

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

3. Simplified 77.9%

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

4. Taylor expanded in K around 0 94.3%

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

   1. cos-neg 94.3%

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

6. Simplified 94.3%

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

7. Taylor expanded in l around inf 34.5%

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

   1. neg-mul-1 34.5%

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

9. Simplified 34.5%

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

10. Final simplification 34.5%

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

Reproduce

herbie shell --seed 2023216 
(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)))))))