Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.9% → 96.5%

Time: 15.3s

Alternatives: 8

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

   1. +-commutative 72.9%

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

   2. +-commutative 72.9%

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

   3. fabs-sub 72.9%

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

   4. associate-/l* 73.5%

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

   5. +-commutative 73.5%

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

3. Simplified 73.5%

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

4. Taylor expanded in K around 0 96.6%

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

   1. cos-neg 96.6%

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

6. Simplified 96.6%

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

7. Final simplification 96.6%

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

Alternative 2: 71.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;n \leq 2.8 \cdot 10^{-86}:\\ \;\;\;\;\cos M \cdot e^{t_0 - \left(\left(m \cdot m\right) \cdot 0.25 + \ell\right)}\\ \mathbf{elif}\;n \leq 4.1 \cdot 10^{+36}:\\ \;\;\;\;\cos \left(\frac{K}{\frac{2}{n}} - M\right) \cdot e^{t_0 - \left(M \cdot M + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t_0 + -0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<= n 2.8e-86)
     (* (cos M) (exp (- t_0 (+ (* (* m m) 0.25) l))))
     (if (<= n 4.1e+36)
       (* (cos (- (/ K (/ 2.0 n)) M)) (exp (- t_0 (+ (* M M) l))))
       (* (cos 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((n - m));
	double tmp;
	if (n <= 2.8e-86) {
		tmp = cos(M) * exp((t_0 - (((m * m) * 0.25) + l)));
	} else if (n <= 4.1e+36) {
		tmp = cos(((K / (2.0 / n)) - M)) * exp((t_0 - ((M * M) + l)));
	} else {
		tmp = cos(M) * 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((n - m))
    if (n <= 2.8d-86) then
        tmp = cos(m_1) * exp((t_0 - (((m * m) * 0.25d0) + l)))
    else if (n <= 4.1d+36) then
        tmp = cos(((k / (2.0d0 / n)) - m_1)) * exp((t_0 - ((m_1 * m_1) + l)))
    else
        tmp = cos(m_1) * 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((n - m));
	double tmp;
	if (n <= 2.8e-86) {
		tmp = Math.cos(M) * Math.exp((t_0 - (((m * m) * 0.25) + l)));
	} else if (n <= 4.1e+36) {
		tmp = Math.cos(((K / (2.0 / n)) - M)) * Math.exp((t_0 - ((M * M) + l)));
	} else {
		tmp = Math.cos(M) * Math.exp((t_0 + (-0.25 * (n * n))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if n <= 2.8e-86:
		tmp = math.cos(M) * math.exp((t_0 - (((m * m) * 0.25) + l)))
	elif n <= 4.1e+36:
		tmp = math.cos(((K / (2.0 / n)) - M)) * math.exp((t_0 - ((M * M) + l)))
	else:
		tmp = math.cos(M) * math.exp((t_0 + (-0.25 * (n * n))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (n <= 2.8e-86)
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(Float64(Float64(m * m) * 0.25) + l))));
	elseif (n <= 4.1e+36)
		tmp = Float64(cos(Float64(Float64(K / Float64(2.0 / n)) - M)) * exp(Float64(t_0 - Float64(Float64(M * M) + l))));
	else
		tmp = Float64(cos(M) * 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((n - m));
	tmp = 0.0;
	if (n <= 2.8e-86)
		tmp = cos(M) * exp((t_0 - (((m * m) * 0.25) + l)));
	elseif (n <= 4.1e+36)
		tmp = cos(((K / (2.0 / n)) - M)) * exp((t_0 - ((M * M) + l)));
	else
		tmp = cos(M) * exp((t_0 + (-0.25 * (n * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, 2.8e-86], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(N[(N[(m * m), $MachinePrecision] * 0.25), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.1e+36], N[(N[Cos[N[(N[(K / N[(2.0 / n), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(N[(M * M), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 + N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < 2.80000000000000009e-86

   1. Initial program 74.3%

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

      1. +-commutative 74.3%

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

      2. +-commutative 74.3%

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

      3. fabs-sub 74.3%

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

      4. associate-/l* 75.1%

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

      5. +-commutative 75.1%

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

   3. Simplified 75.1%

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

   4. Taylor expanded in K around 0 95.5%

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

      1. cos-neg 95.5%

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

   6. Simplified 95.5%

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

   7. Taylor expanded in m around inf 65.4%

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

      1. *-commutative 65.4%

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

      2. unpow2 65.4%

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

   9. Simplified 65.4%

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

   if 2.80000000000000009e-86 < n < 4.10000000000000013e36

   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%

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

      2. +-commutative 80.5%

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

      3. fabs-sub 80.5%

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

      4. associate-/l* 80.5%

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

      5. +-commutative 80.5%

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

   3. Simplified 80.5%

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

   4. Taylor expanded in M around inf 60.9%

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

      1. unpow2 60.9%

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

   6. Simplified 60.9%

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

   7. Taylor expanded in m around 0 77.5%

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

   if 4.10000000000000013e36 < n

   1. Initial program 64.2%

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

      1. +-commutative 64.2%

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

      2. +-commutative 64.2%

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

      3. fabs-sub 64.2%

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

      4. associate-/l* 64.2%

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

      5. +-commutative 64.2%

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

   3. Simplified 64.2%

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

   4. Taylor expanded in n around 0 47.2%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-\color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + \left(n \cdot \left(0.5 \cdot m - M\right) + 0.25 \cdot {n}^{2}\right)\right)}\right) - \ell\right) + \left|n - m\right|} \]
   5. Step-by-step derivation

      1. associate-+r+ 47.2%

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

      2. unpow2 47.2%

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

      3. distribute-rgt-out 62.3%

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

      4. *-commutative 62.3%

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

      5. *-commutative 62.3%

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

      6. unpow2 62.3%

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

      7. associate-*r* 62.3%

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

   6. Simplified 62.3%

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

   7. Taylor expanded in m around 0 62.3%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-\color{blue}{\left(-1 \cdot \left(\left(n - M\right) \cdot M\right) + 0.25 \cdot {n}^{2}\right)}\right) - \ell\right) + \left|n - m\right|} \]
   8. Step-by-step derivation

      1. +-commutative 62.3%

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

      2. *-commutative 62.3%

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

      3. unpow2 62.3%

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

      4. associate-*r* 62.3%

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

      5. mul-1-neg 62.3%

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

      6. unsub-neg 62.3%

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

      7. *-commutative 62.3%

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

   9. Simplified 62.3%

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

   10. Taylor expanded in K around 0 92.5%

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

       1. cos-neg 100.0%

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

   12. Simplified 92.5%

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

   13. Taylor expanded in n around inf 94.4%

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

       1. unpow2 94.4%

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

   15. Simplified 94.4%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2.8 \cdot 10^{-86}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - \left(\left(m \cdot m\right) \cdot 0.25 + \ell\right)}\\ \mathbf{elif}\;n \leq 4.1 \cdot 10^{+36}:\\ \;\;\;\;\cos \left(\frac{K}{\frac{2}{n}} - M\right) \cdot e^{\left|n - m\right| - \left(M \cdot M + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| + -0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]

Alternative 3: 83.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;m \leq -52000000000:\\ \;\;\;\;\cos M \cdot e^{t_0 - \left(\left(m \cdot m\right) \cdot 0.25 + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t_0 + \left(\left(M \cdot \left(n - M\right) - n \cdot \left(n \cdot 0.25\right)\right) - \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<= m -52000000000.0)
     (* (cos M) (exp (- t_0 (+ (* (* m m) 0.25) l))))
     (* (cos M) (exp (+ t_0 (- (- (* M (- n M)) (* n (* n 0.25))) l)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if (m <= -52000000000.0) {
		tmp = cos(M) * exp((t_0 - (((m * m) * 0.25) + l)));
	} else {
		tmp = cos(M) * exp((t_0 + (((M * (n - M)) - (n * (n * 0.25))) - l)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((n - m))
    if (m <= (-52000000000.0d0)) then
        tmp = cos(m_1) * exp((t_0 - (((m * m) * 0.25d0) + l)))
    else
        tmp = cos(m_1) * exp((t_0 + (((m_1 * (n - m_1)) - (n * (n * 0.25d0))) - l)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((n - m));
	double tmp;
	if (m <= -52000000000.0) {
		tmp = Math.cos(M) * Math.exp((t_0 - (((m * m) * 0.25) + l)));
	} else {
		tmp = Math.cos(M) * Math.exp((t_0 + (((M * (n - M)) - (n * (n * 0.25))) - l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if m <= -52000000000.0:
		tmp = math.cos(M) * math.exp((t_0 - (((m * m) * 0.25) + l)))
	else:
		tmp = math.cos(M) * math.exp((t_0 + (((M * (n - M)) - (n * (n * 0.25))) - l)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (m <= -52000000000.0)
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(Float64(Float64(m * m) * 0.25) + l))));
	else
		tmp = Float64(cos(M) * exp(Float64(t_0 + Float64(Float64(Float64(M * Float64(n - M)) - Float64(n * Float64(n * 0.25))) - l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m));
	tmp = 0.0;
	if (m <= -52000000000.0)
		tmp = cos(M) * exp((t_0 - (((m * m) * 0.25) + l)));
	else
		tmp = cos(M) * exp((t_0 + (((M * (n - M)) - (n * (n * 0.25))) - l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[m, -52000000000.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(N[(N[(m * m), $MachinePrecision] * 0.25), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 + N[(N[(N[(M * N[(n - M), $MachinePrecision]), $MachinePrecision] - N[(n * N[(n * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < -5.2e10

   1. Initial program 54.2%

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

      1. +-commutative 54.2%

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

      2. +-commutative 54.2%

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

      3. fabs-sub 54.2%

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

      4. associate-/l* 54.2%

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

      5. +-commutative 54.2%

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

   3. Simplified 54.2%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in m around inf 86.7%

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

      1. *-commutative 86.7%

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

      2. unpow2 86.7%

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

   9. Simplified 86.7%

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

   if -5.2e10 < m

   1. Initial program 78.5%

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

      1. +-commutative 78.5%

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

      2. +-commutative 78.5%

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

      3. fabs-sub 78.5%

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

      4. associate-/l* 79.2%

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

      5. +-commutative 79.2%

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

   3. Simplified 79.2%

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

   4. Taylor expanded in n around 0 73.1%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-\color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + \left(n \cdot \left(0.5 \cdot m - M\right) + 0.25 \cdot {n}^{2}\right)\right)}\right) - \ell\right) + \left|n - m\right|} \]
   5. Step-by-step derivation

      1. associate-+r+ 73.1%

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

      2. unpow2 73.1%

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

      3. distribute-rgt-out 78.2%

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

      4. *-commutative 78.2%

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

      5. *-commutative 78.2%

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

      6. unpow2 78.2%

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

      7. associate-*r* 78.2%

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

   6. Simplified 78.2%

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

   7. Taylor expanded in m around 0 73.4%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-\color{blue}{\left(-1 \cdot \left(\left(n - M\right) \cdot M\right) + 0.25 \cdot {n}^{2}\right)}\right) - \ell\right) + \left|n - m\right|} \]
   8. Step-by-step derivation

      1. +-commutative 73.4%

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

      2. *-commutative 73.4%

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

      3. unpow2 73.4%

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

      4. associate-*r* 73.4%

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

      5. mul-1-neg 73.4%

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

      6. unsub-neg 73.4%

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

      7. *-commutative 73.4%

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

   9. Simplified 73.4%

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

   10. Taylor expanded in K around 0 84.8%

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

       1. cos-neg 95.6%

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

   12. Simplified 84.8%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -52000000000:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - \left(\left(m \cdot m\right) \cdot 0.25 + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| + \left(\left(M \cdot \left(n - M\right) - n \cdot \left(n \cdot 0.25\right)\right) - \ell\right)}\\ \end{array} \]

Alternative 4: 71.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;n \leq 1.4 \cdot 10^{-86}:\\ \;\;\;\;\cos M \cdot e^{t_0 - \left(\left(m \cdot m\right) \cdot 0.25 + \ell\right)}\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{+38}:\\ \;\;\;\;\cos M \cdot e^{t_0 - \left(M \cdot M + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t_0 + -0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<= n 1.4e-86)
     (* (cos M) (exp (- t_0 (+ (* (* m m) 0.25) l))))
     (if (<= n 1.5e+38)
       (* (cos M) (exp (- t_0 (+ (* M M) l))))
       (* (cos 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((n - m));
	double tmp;
	if (n <= 1.4e-86) {
		tmp = cos(M) * exp((t_0 - (((m * m) * 0.25) + l)));
	} else if (n <= 1.5e+38) {
		tmp = cos(M) * exp((t_0 - ((M * M) + l)));
	} else {
		tmp = cos(M) * 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((n - m))
    if (n <= 1.4d-86) then
        tmp = cos(m_1) * exp((t_0 - (((m * m) * 0.25d0) + l)))
    else if (n <= 1.5d+38) then
        tmp = cos(m_1) * exp((t_0 - ((m_1 * m_1) + l)))
    else
        tmp = cos(m_1) * exp((t_0 + ((-0.25d0) * (n * n))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((n - m));
	double tmp;
	if (n <= 1.4e-86) {
		tmp = Math.cos(M) * Math.exp((t_0 - (((m * m) * 0.25) + l)));
	} else if (n <= 1.5e+38) {
		tmp = Math.cos(M) * Math.exp((t_0 - ((M * M) + l)));
	} else {
		tmp = Math.cos(M) * Math.exp((t_0 + (-0.25 * (n * n))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if n <= 1.4e-86:
		tmp = math.cos(M) * math.exp((t_0 - (((m * m) * 0.25) + l)))
	elif n <= 1.5e+38:
		tmp = math.cos(M) * math.exp((t_0 - ((M * M) + l)))
	else:
		tmp = math.cos(M) * math.exp((t_0 + (-0.25 * (n * n))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (n <= 1.4e-86)
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(Float64(Float64(m * m) * 0.25) + l))));
	elseif (n <= 1.5e+38)
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(Float64(M * M) + l))));
	else
		tmp = Float64(cos(M) * 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((n - m));
	tmp = 0.0;
	if (n <= 1.4e-86)
		tmp = cos(M) * exp((t_0 - (((m * m) * 0.25) + l)));
	elseif (n <= 1.5e+38)
		tmp = cos(M) * exp((t_0 - ((M * M) + l)));
	else
		tmp = cos(M) * exp((t_0 + (-0.25 * (n * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, 1.4e-86], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(N[(N[(m * m), $MachinePrecision] * 0.25), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.5e+38], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(N[(M * M), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 + N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < 1.40000000000000005e-86

   1. Initial program 74.6%

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

      1. +-commutative 74.6%

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

      2. +-commutative 74.6%

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

      3. fabs-sub 74.6%

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

      4. associate-/l* 75.4%

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

      5. +-commutative 75.4%

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

   3. Simplified 75.4%

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

   4. Taylor expanded in K around 0 95.9%

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

      1. cos-neg 95.9%

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

   6. Simplified 95.9%

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

   7. Taylor expanded in m around inf 65.6%

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

      1. *-commutative 65.6%

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

      2. unpow2 65.6%

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

   9. Simplified 65.6%

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

   if 1.40000000000000005e-86 < n < 1.5000000000000001e38

   1. Initial program 78.6%

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

      1. +-commutative 78.6%

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

      2. +-commutative 78.6%

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

      3. fabs-sub 78.6%

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

      4. associate-/l* 78.6%

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

      5. +-commutative 78.6%

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

   3. Simplified 78.6%

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

   4. Taylor expanded in K around 0 95.3%

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

      1. cos-neg 95.3%

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

   6. Simplified 95.3%

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

   7. Taylor expanded in M around inf 76.2%

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

      1. unpow2 59.6%

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

   9. Simplified 76.2%

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

   if 1.5000000000000001e38 < n

   1. Initial program 64.2%

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

      1. +-commutative 64.2%

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

      2. +-commutative 64.2%

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

      3. fabs-sub 64.2%

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

      4. associate-/l* 64.2%

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

      5. +-commutative 64.2%

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

   3. Simplified 64.2%

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

   4. Taylor expanded in n around 0 47.2%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-\color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + \left(n \cdot \left(0.5 \cdot m - M\right) + 0.25 \cdot {n}^{2}\right)\right)}\right) - \ell\right) + \left|n - m\right|} \]
   5. Step-by-step derivation

      1. associate-+r+ 47.2%

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

      2. unpow2 47.2%

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

      3. distribute-rgt-out 62.3%

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

      4. *-commutative 62.3%

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

      5. *-commutative 62.3%

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

      6. unpow2 62.3%

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

      7. associate-*r* 62.3%

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

   6. Simplified 62.3%

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

   7. Taylor expanded in m around 0 62.3%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-\color{blue}{\left(-1 \cdot \left(\left(n - M\right) \cdot M\right) + 0.25 \cdot {n}^{2}\right)}\right) - \ell\right) + \left|n - m\right|} \]
   8. Step-by-step derivation

      1. +-commutative 62.3%

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

      2. *-commutative 62.3%

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

      3. unpow2 62.3%

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

      4. associate-*r* 62.3%

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

      5. mul-1-neg 62.3%

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

      6. unsub-neg 62.3%

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

      7. *-commutative 62.3%

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

   9. Simplified 62.3%

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

   10. Taylor expanded in K around 0 92.5%

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

       1. cos-neg 100.0%

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

   12. Simplified 92.5%

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

   13. Taylor expanded in n around inf 94.4%

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

       1. unpow2 94.4%

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

   15. Simplified 94.4%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 1.4 \cdot 10^{-86}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - \left(\left(m \cdot m\right) \cdot 0.25 + \ell\right)}\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{+38}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - \left(M \cdot M + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| + -0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]

Alternative 5: 55.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;M \leq -5.8 \cdot 10^{+37} \lor \neg \left(M \leq 3.8 \cdot 10^{+25}\right):\\ \;\;\;\;\cos M \cdot e^{t_0 - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t_0 - \ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (or (<= M -5.8e+37) (not (<= M 3.8e+25)))
     (* (cos M) (exp (- t_0 (* M M))))
     (* (cos M) (exp (- t_0 l))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double tmp;
	if ((M <= -5.8e+37) || !(M <= 3.8e+25)) {
		tmp = cos(M) * exp((t_0 - (M * M)));
	} else {
		tmp = cos(M) * exp((t_0 - l));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((n - m))
    if ((m_1 <= (-5.8d+37)) .or. (.not. (m_1 <= 3.8d+25))) then
        tmp = cos(m_1) * exp((t_0 - (m_1 * m_1)))
    else
        tmp = cos(m_1) * exp((t_0 - l))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((n - m));
	double tmp;
	if ((M <= -5.8e+37) || !(M <= 3.8e+25)) {
		tmp = Math.cos(M) * Math.exp((t_0 - (M * M)));
	} else {
		tmp = Math.cos(M) * Math.exp((t_0 - l));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if (M <= -5.8e+37) or not (M <= 3.8e+25):
		tmp = math.cos(M) * math.exp((t_0 - (M * M)))
	else:
		tmp = math.cos(M) * math.exp((t_0 - l))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if ((M <= -5.8e+37) || !(M <= 3.8e+25))
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(M * M))));
	else
		tmp = Float64(cos(M) * exp(Float64(t_0 - l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m));
	tmp = 0.0;
	if ((M <= -5.8e+37) || ~((M <= 3.8e+25)))
		tmp = cos(M) * exp((t_0 - (M * M)));
	else
		tmp = cos(M) * exp((t_0 - l));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[M, -5.8e+37], N[Not[LessEqual[M, 3.8e+25]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < -5.79999999999999957e37 or 3.8e25 < M

   1. Initial program 80.8%

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

      1. +-commutative 80.8%

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

      2. +-commutative 80.8%

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

      3. fabs-sub 80.8%

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

      4. associate-/l* 80.8%

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

      5. +-commutative 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in K around 0 99.2%

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

      1. cos-neg 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in M around inf 88.5%

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

      1. unpow2 76.7%

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

   9. Simplified 88.5%

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

   10. Taylor expanded in M around inf 89.3%

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

       1. mul-1-neg 89.3%

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

       2. unpow2 89.3%

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

       3. distribute-rgt-neg-out 89.3%

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

   12. Simplified 89.3%

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

   if -5.79999999999999957e37 < M < 3.8e25

   1. Initial program 66.0%

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

      1. +-commutative 66.0%

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

      2. +-commutative 66.0%

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

      3. fabs-sub 66.0%

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

      4. associate-/l* 67.0%

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

      5. +-commutative 67.0%

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

   3. Simplified 67.0%

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

   4. Taylor expanded in M around inf 27.4%

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

      1. unpow2 27.4%

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

   6. Simplified 27.4%

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

   7. Taylor expanded in M around 0 26.7%

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

      1. neg-mul-1 26.7%

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

   9. Simplified 26.7%

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

   10. Taylor expanded in K around 0 29.5%

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

       1. cos-neg 94.4%

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

   12. Simplified 29.5%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -5.8 \cdot 10^{+37} \lor \neg \left(M \leq 3.8 \cdot 10^{+25}\right):\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - \ell}\\ \end{array} \]

Alternative 6: 57.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;n \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\cos M \cdot e^{t_0 - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t_0 + -0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<= n 5e+36)
     (* (cos M) (exp (- t_0 (* M M))))
     (* (cos 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((n - m));
	double tmp;
	if (n <= 5e+36) {
		tmp = cos(M) * exp((t_0 - (M * M)));
	} else {
		tmp = cos(M) * 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((n - m))
    if (n <= 5d+36) then
        tmp = cos(m_1) * exp((t_0 - (m_1 * m_1)))
    else
        tmp = cos(m_1) * 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((n - m));
	double tmp;
	if (n <= 5e+36) {
		tmp = Math.cos(M) * Math.exp((t_0 - (M * M)));
	} else {
		tmp = Math.cos(M) * Math.exp((t_0 + (-0.25 * (n * n))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if n <= 5e+36:
		tmp = math.cos(M) * math.exp((t_0 - (M * M)))
	else:
		tmp = math.cos(M) * math.exp((t_0 + (-0.25 * (n * n))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (n <= 5e+36)
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(M * M))));
	else
		tmp = Float64(cos(M) * 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((n - m));
	tmp = 0.0;
	if (n <= 5e+36)
		tmp = cos(M) * exp((t_0 - (M * M)));
	else
		tmp = cos(M) * exp((t_0 + (-0.25 * (n * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, 5e+36], 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[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < 4.99999999999999977e36

   1. Initial program 75.2%

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

      1. +-commutative 75.2%

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

      2. +-commutative 75.2%

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

      3. fabs-sub 75.2%

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

      4. associate-/l* 75.9%

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

      5. +-commutative 75.9%

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

   3. Simplified 75.9%

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

   4. Taylor expanded in K around 0 95.8%

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

      1. cos-neg 95.8%

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

   6. Simplified 95.8%

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

   7. Taylor expanded in M around inf 61.0%

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

      1. unpow2 54.7%

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

   9. Simplified 61.0%

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

   10. Taylor expanded in M around inf 49.9%

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

       1. mul-1-neg 49.9%

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

       2. unpow2 49.9%

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

       3. distribute-rgt-neg-out 49.9%

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

   12. Simplified 49.9%

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

   if 4.99999999999999977e36 < n

   1. Initial program 64.2%

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

      1. +-commutative 64.2%

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

      2. +-commutative 64.2%

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

      3. fabs-sub 64.2%

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

      4. associate-/l* 64.2%

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

      5. +-commutative 64.2%

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

   3. Simplified 64.2%

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

   4. Taylor expanded in n around 0 47.2%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-\color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + \left(n \cdot \left(0.5 \cdot m - M\right) + 0.25 \cdot {n}^{2}\right)\right)}\right) - \ell\right) + \left|n - m\right|} \]
   5. Step-by-step derivation

      1. associate-+r+ 47.2%

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

      2. unpow2 47.2%

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

      3. distribute-rgt-out 62.3%

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

      4. *-commutative 62.3%

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

      5. *-commutative 62.3%

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

      6. unpow2 62.3%

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

      7. associate-*r* 62.3%

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

   6. Simplified 62.3%

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

   7. Taylor expanded in m around 0 62.3%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-\color{blue}{\left(-1 \cdot \left(\left(n - M\right) \cdot M\right) + 0.25 \cdot {n}^{2}\right)}\right) - \ell\right) + \left|n - m\right|} \]
   8. Step-by-step derivation

      1. +-commutative 62.3%

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

      2. *-commutative 62.3%

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

      3. unpow2 62.3%

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

      4. associate-*r* 62.3%

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

      5. mul-1-neg 62.3%

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

      6. unsub-neg 62.3%

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

      7. *-commutative 62.3%

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

   9. Simplified 62.3%

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

   10. Taylor expanded in K around 0 92.5%

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

       1. cos-neg 100.0%

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

   12. Simplified 92.5%

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

   13. Taylor expanded in n around inf 94.4%

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

       1. unpow2 94.4%

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

   15. Simplified 94.4%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| + -0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]

Alternative 7: 67.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ \mathbf{if}\;n \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\cos M \cdot e^{t_0 - \left(M \cdot M + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t_0 + -0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m))))
   (if (<= n 5e+36)
     (* (cos M) (exp (- t_0 (+ (* M M) l))))
     (* (cos 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((n - m));
	double tmp;
	if (n <= 5e+36) {
		tmp = cos(M) * exp((t_0 - ((M * M) + l)));
	} else {
		tmp = cos(M) * 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((n - m))
    if (n <= 5d+36) then
        tmp = cos(m_1) * exp((t_0 - ((m_1 * m_1) + l)))
    else
        tmp = cos(m_1) * exp((t_0 + ((-0.25d0) * (n * n))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((n - m));
	double tmp;
	if (n <= 5e+36) {
		tmp = Math.cos(M) * Math.exp((t_0 - ((M * M) + l)));
	} else {
		tmp = Math.cos(M) * Math.exp((t_0 + (-0.25 * (n * n))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	tmp = 0
	if n <= 5e+36:
		tmp = math.cos(M) * math.exp((t_0 - ((M * M) + l)))
	else:
		tmp = math.cos(M) * math.exp((t_0 + (-0.25 * (n * n))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	tmp = 0.0
	if (n <= 5e+36)
		tmp = Float64(cos(M) * exp(Float64(t_0 - Float64(Float64(M * M) + l))));
	else
		tmp = Float64(cos(M) * 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((n - m));
	tmp = 0.0;
	if (n <= 5e+36)
		tmp = cos(M) * exp((t_0 - ((M * M) + l)));
	else
		tmp = cos(M) * exp((t_0 + (-0.25 * (n * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, 5e+36], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 - N[(N[(M * M), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(t$95$0 + N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < 4.99999999999999977e36

   1. Initial program 75.2%

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

      1. +-commutative 75.2%

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

      2. +-commutative 75.2%

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

      3. fabs-sub 75.2%

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

      4. associate-/l* 75.9%

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

      5. +-commutative 75.9%

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

   3. Simplified 75.9%

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

   4. Taylor expanded in K around 0 95.8%

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

      1. cos-neg 95.8%

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

   6. Simplified 95.8%

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

   7. Taylor expanded in M around inf 61.0%

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

      1. unpow2 54.7%

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

   9. Simplified 61.0%

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

   if 4.99999999999999977e36 < n

   1. Initial program 64.2%

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

      1. +-commutative 64.2%

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

      2. +-commutative 64.2%

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

      3. fabs-sub 64.2%

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

      4. associate-/l* 64.2%

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

      5. +-commutative 64.2%

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

   3. Simplified 64.2%

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

   4. Taylor expanded in n around 0 47.2%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-\color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + \left(n \cdot \left(0.5 \cdot m - M\right) + 0.25 \cdot {n}^{2}\right)\right)}\right) - \ell\right) + \left|n - m\right|} \]
   5. Step-by-step derivation

      1. associate-+r+ 47.2%

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

      2. unpow2 47.2%

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

      3. distribute-rgt-out 62.3%

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

      4. *-commutative 62.3%

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

      5. *-commutative 62.3%

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

      6. unpow2 62.3%

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

      7. associate-*r* 62.3%

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

   6. Simplified 62.3%

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

   7. Taylor expanded in m around 0 62.3%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-\color{blue}{\left(-1 \cdot \left(\left(n - M\right) \cdot M\right) + 0.25 \cdot {n}^{2}\right)}\right) - \ell\right) + \left|n - m\right|} \]
   8. Step-by-step derivation

      1. +-commutative 62.3%

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

      2. *-commutative 62.3%

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

      3. unpow2 62.3%

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

      4. associate-*r* 62.3%

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

      5. mul-1-neg 62.3%

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

      6. unsub-neg 62.3%

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

      7. *-commutative 62.3%

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

   9. Simplified 62.3%

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

   10. Taylor expanded in K around 0 92.5%

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

       1. cos-neg 100.0%

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

   12. Simplified 92.5%

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

   13. Taylor expanded in n around inf 94.4%

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

       1. unpow2 94.4%

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

   15. Simplified 94.4%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| - \left(M \cdot M + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| + -0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]

Alternative 8: 25.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

   1. +-commutative 72.9%

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

   2. +-commutative 72.9%

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

   3. fabs-sub 72.9%

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

   4. associate-/l* 73.5%

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

   5. +-commutative 73.5%

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

3. Simplified 73.5%

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

4. Taylor expanded in M around inf 50.5%

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

   1. unpow2 50.5%

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

6. Simplified 50.5%

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

7. Taylor expanded in M around 0 21.0%

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

   1. neg-mul-1 21.0%

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

9. Simplified 21.0%

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

10. Taylor expanded in K around 0 22.7%

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

    1. cos-neg 96.6%

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

12. Simplified 22.7%

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

13. Final simplification 22.7%

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

Reproduce

herbie shell --seed 2023272 
(FPCore (K m n M l)
  :name "Maksimov and Kolovsky, Equation (32)"
  :precision binary64
  (* (cos (- (/ (* K (+ m n)) 2.0) M)) (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))