Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.8% → 96.7%

Time: 21.9s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.8% 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: 96.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

3. Simplified 74.3%

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

4. Taylor expanded in K around 0 98.4%

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

   1. neg-mul-1 98.4%

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

6. Simplified 98.4%

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

7. Final simplification 98.4%

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

Alternative 2: 73.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|n - m\right|\\ t_1 := \frac{\cos M}{e^{\left(\ell - t_0\right) + M \cdot M}}\\ t_2 := e^{t_0 - \left(\ell + 0.25 \cdot \left(m \cdot m\right)\right)}\\ \mathbf{if}\;n \leq -1.02 \cdot 10^{-179}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;n \leq -3.1 \cdot 10^{-257}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-126}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;n \leq 52:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- n m)))
        (t_1 (/ (cos M) (exp (+ (- l t_0) (* M M)))))
        (t_2 (exp (- t_0 (+ l (* 0.25 (* m m)))))))
   (if (<= n -1.02e-179)
     t_2
     (if (<= n -3.1e-257)
       t_1
       (if (<= n 1.15e-126)
         t_2
         (if (<= n 52.0) t_1 (/ (cos M) (exp (* 0.25 (* n n))))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((n - m));
	double t_1 = cos(M) / exp(((l - t_0) + (M * M)));
	double t_2 = exp((t_0 - (l + (0.25 * (m * m)))));
	double tmp;
	if (n <= -1.02e-179) {
		tmp = t_2;
	} else if (n <= -3.1e-257) {
		tmp = t_1;
	} else if (n <= 1.15e-126) {
		tmp = t_2;
	} else if (n <= 52.0) {
		tmp = t_1;
	} else {
		tmp = cos(M) / exp((0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = abs((n - m))
    t_1 = cos(m_1) / exp(((l - t_0) + (m_1 * m_1)))
    t_2 = exp((t_0 - (l + (0.25d0 * (m * m)))))
    if (n <= (-1.02d-179)) then
        tmp = t_2
    else if (n <= (-3.1d-257)) then
        tmp = t_1
    else if (n <= 1.15d-126) then
        tmp = t_2
    else if (n <= 52.0d0) then
        tmp = t_1
    else
        tmp = cos(m_1) / exp((0.25d0 * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((n - m));
	double t_1 = Math.cos(M) / Math.exp(((l - t_0) + (M * M)));
	double t_2 = Math.exp((t_0 - (l + (0.25 * (m * m)))));
	double tmp;
	if (n <= -1.02e-179) {
		tmp = t_2;
	} else if (n <= -3.1e-257) {
		tmp = t_1;
	} else if (n <= 1.15e-126) {
		tmp = t_2;
	} else if (n <= 52.0) {
		tmp = t_1;
	} else {
		tmp = Math.cos(M) / Math.exp((0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.fabs((n - m))
	t_1 = math.cos(M) / math.exp(((l - t_0) + (M * M)))
	t_2 = math.exp((t_0 - (l + (0.25 * (m * m)))))
	tmp = 0
	if n <= -1.02e-179:
		tmp = t_2
	elif n <= -3.1e-257:
		tmp = t_1
	elif n <= 1.15e-126:
		tmp = t_2
	elif n <= 52.0:
		tmp = t_1
	else:
		tmp = math.cos(M) / math.exp((0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = abs(Float64(n - m))
	t_1 = Float64(cos(M) / exp(Float64(Float64(l - t_0) + Float64(M * M))))
	t_2 = exp(Float64(t_0 - Float64(l + Float64(0.25 * Float64(m * m)))))
	tmp = 0.0
	if (n <= -1.02e-179)
		tmp = t_2;
	elseif (n <= -3.1e-257)
		tmp = t_1;
	elseif (n <= 1.15e-126)
		tmp = t_2;
	elseif (n <= 52.0)
		tmp = t_1;
	else
		tmp = Float64(cos(M) / exp(Float64(0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((n - m));
	t_1 = cos(M) / exp(((l - t_0) + (M * M)));
	t_2 = exp((t_0 - (l + (0.25 * (m * m)))));
	tmp = 0.0;
	if (n <= -1.02e-179)
		tmp = t_2;
	elseif (n <= -3.1e-257)
		tmp = t_1;
	elseif (n <= 1.15e-126)
		tmp = t_2;
	elseif (n <= 52.0)
		tmp = t_1;
	else
		tmp = cos(M) / exp((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]}, Block[{t$95$1 = N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(N[(l - t$95$0), $MachinePrecision] + N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(t$95$0 - N[(l + N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -1.02e-179], t$95$2, If[LessEqual[n, -3.1e-257], t$95$1, If[LessEqual[n, 1.15e-126], t$95$2, If[LessEqual[n, 52.0], t$95$1, N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.02e-179 or -3.10000000000000008e-257 < n < 1.15000000000000005e-126

   1. Initial program 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in m around inf 55.5%

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

      1. *-commutative 55.5%

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

      2. unpow2 55.5%

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

   6. Simplified 55.5%

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

   7. Taylor expanded in K around 0 68.0%

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

      1. cos-neg 59.1%

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

   9. Simplified 68.0%

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

   10. Taylor expanded in M around 0 67.3%

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

       1. sub-neg 67.3%

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

       2. mul-1-neg 67.3%

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

       3. +-commutative 67.3%

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

       4. mul-1-neg 67.3%

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

       5. sub-neg 67.3%

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

       6. associate-+r- 67.3%

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

       7. *-commutative 67.3%

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

       8. fma-def 67.3%

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

       9. unpow2 67.3%

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

       10. rem-log-exp 67.3%

           \[\leadsto \frac{1}{e^{\color{blue}{\log \left(e^{\mathsf{fma}\left(m \cdot m, 0.25, \ell - \left|n - m\right|\right)}\right)}}} \]

       11. rec-exp 67.3%

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

       12. rem-log-exp 67.3%

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

   12. Simplified 67.3%

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

   if -1.02e-179 < n < -3.10000000000000008e-257 or 1.15000000000000005e-126 < n < 52

   1. Initial program 76.7%

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

   3. Simplified 76.7%

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

   4. Taylor expanded in M around inf 67.6%

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

      1. unpow2 67.6%

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

   6. Simplified 67.6%

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

   7. Taylor expanded in K around 0 78.5%

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

      1. cos-neg 24.0%

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

   9. Simplified 78.5%

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

   if 52 < n

   1. Initial program 68.1%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in n around inf 58.1%

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

      1. *-commutative 58.1%

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

      2. unpow2 58.1%

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

   6. Simplified 58.1%

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

   7. Taylor expanded in K around 0 82.9%

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

      1. cos-neg 82.9%

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

   9. Simplified 82.9%

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

   10. Taylor expanded in n around inf 98.6%

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

       1. unpow2 98.6%

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

   12. Simplified 98.6%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.02 \cdot 10^{-179}:\\ \;\;\;\;e^{\left|n - m\right| - \left(\ell + 0.25 \cdot \left(m \cdot m\right)\right)}\\ \mathbf{elif}\;n \leq -3.1 \cdot 10^{-257}:\\ \;\;\;\;\frac{\cos M}{e^{\left(\ell - \left|n - m\right|\right) + M \cdot M}}\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-126}:\\ \;\;\;\;e^{\left|n - m\right| - \left(\ell + 0.25 \cdot \left(m \cdot m\right)\right)}\\ \mathbf{elif}\;n \leq 52:\\ \;\;\;\;\frac{\cos M}{e^{\left(\ell - \left|n - m\right|\right) + M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \]

Alternative 3: 73.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.25 \cdot \left(m \cdot m\right)\\ t_1 := \left|n - m\right|\\ t_2 := \ell - t_1\\ t_3 := \frac{\cos M}{e^{t_2 + M \cdot M}}\\ \mathbf{if}\;n \leq -1.35 \cdot 10^{-179}:\\ \;\;\;\;e^{t_1 - \left(\ell + t_0\right)}\\ \mathbf{elif}\;n \leq -9.5 \cdot 10^{-257}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{-126}:\\ \;\;\;\;\frac{\cos M}{e^{t_2 + t_0}}\\ \mathbf{elif}\;n \leq 52:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* 0.25 (* m m)))
        (t_1 (fabs (- n m)))
        (t_2 (- l t_1))
        (t_3 (/ (cos M) (exp (+ t_2 (* M M))))))
   (if (<= n -1.35e-179)
     (exp (- t_1 (+ l t_0)))
     (if (<= n -9.5e-257)
       t_3
       (if (<= n 1.45e-126)
         (/ (cos M) (exp (+ t_2 t_0)))
         (if (<= n 52.0) t_3 (/ (cos M) (exp (* 0.25 (* n n))))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = 0.25 * (m * m);
	double t_1 = fabs((n - m));
	double t_2 = l - t_1;
	double t_3 = cos(M) / exp((t_2 + (M * M)));
	double tmp;
	if (n <= -1.35e-179) {
		tmp = exp((t_1 - (l + t_0)));
	} else if (n <= -9.5e-257) {
		tmp = t_3;
	} else if (n <= 1.45e-126) {
		tmp = cos(M) / exp((t_2 + t_0));
	} else if (n <= 52.0) {
		tmp = t_3;
	} else {
		tmp = cos(M) / exp((0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 0.25d0 * (m * m)
    t_1 = abs((n - m))
    t_2 = l - t_1
    t_3 = cos(m_1) / exp((t_2 + (m_1 * m_1)))
    if (n <= (-1.35d-179)) then
        tmp = exp((t_1 - (l + t_0)))
    else if (n <= (-9.5d-257)) then
        tmp = t_3
    else if (n <= 1.45d-126) then
        tmp = cos(m_1) / exp((t_2 + t_0))
    else if (n <= 52.0d0) then
        tmp = t_3
    else
        tmp = cos(m_1) / exp((0.25d0 * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = 0.25 * (m * m);
	double t_1 = Math.abs((n - m));
	double t_2 = l - t_1;
	double t_3 = Math.cos(M) / Math.exp((t_2 + (M * M)));
	double tmp;
	if (n <= -1.35e-179) {
		tmp = Math.exp((t_1 - (l + t_0)));
	} else if (n <= -9.5e-257) {
		tmp = t_3;
	} else if (n <= 1.45e-126) {
		tmp = Math.cos(M) / Math.exp((t_2 + t_0));
	} else if (n <= 52.0) {
		tmp = t_3;
	} else {
		tmp = Math.cos(M) / Math.exp((0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = 0.25 * (m * m)
	t_1 = math.fabs((n - m))
	t_2 = l - t_1
	t_3 = math.cos(M) / math.exp((t_2 + (M * M)))
	tmp = 0
	if n <= -1.35e-179:
		tmp = math.exp((t_1 - (l + t_0)))
	elif n <= -9.5e-257:
		tmp = t_3
	elif n <= 1.45e-126:
		tmp = math.cos(M) / math.exp((t_2 + t_0))
	elif n <= 52.0:
		tmp = t_3
	else:
		tmp = math.cos(M) / math.exp((0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(0.25 * Float64(m * m))
	t_1 = abs(Float64(n - m))
	t_2 = Float64(l - t_1)
	t_3 = Float64(cos(M) / exp(Float64(t_2 + Float64(M * M))))
	tmp = 0.0
	if (n <= -1.35e-179)
		tmp = exp(Float64(t_1 - Float64(l + t_0)));
	elseif (n <= -9.5e-257)
		tmp = t_3;
	elseif (n <= 1.45e-126)
		tmp = Float64(cos(M) / exp(Float64(t_2 + t_0)));
	elseif (n <= 52.0)
		tmp = t_3;
	else
		tmp = Float64(cos(M) / exp(Float64(0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = 0.25 * (m * m);
	t_1 = abs((n - m));
	t_2 = l - t_1;
	t_3 = cos(M) / exp((t_2 + (M * M)));
	tmp = 0.0;
	if (n <= -1.35e-179)
		tmp = exp((t_1 - (l + t_0)));
	elseif (n <= -9.5e-257)
		tmp = t_3;
	elseif (n <= 1.45e-126)
		tmp = cos(M) / exp((t_2 + t_0));
	elseif (n <= 52.0)
		tmp = t_3;
	else
		tmp = cos(M) / exp((0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(l - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(t$95$2 + N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.35e-179], N[Exp[N[(t$95$1 - N[(l + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, -9.5e-257], t$95$3, If[LessEqual[n, 1.45e-126], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(t$95$2 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 52.0], t$95$3, N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.34999999999999994e-179

   1. Initial program 73.2%

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

   3. Simplified 73.2%

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

   4. Taylor expanded in m around inf 45.6%

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

      1. *-commutative 45.6%

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

      2. unpow2 45.6%

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

   6. Simplified 45.6%

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

   7. Taylor expanded in K around 0 57.3%

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

      1. cos-neg 70.6%

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

   9. Simplified 57.3%

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

   10. Taylor expanded in M around 0 56.2%

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

       1. sub-neg 56.2%

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

       2. mul-1-neg 56.2%

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

       3. +-commutative 56.2%

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

       4. mul-1-neg 56.2%

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

       5. sub-neg 56.2%

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

       6. associate-+r- 56.2%

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

       7. *-commutative 56.2%

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

       8. fma-def 56.2%

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

       9. unpow2 56.2%

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

       10. rem-log-exp 56.2%

           \[\leadsto \frac{1}{e^{\color{blue}{\log \left(e^{\mathsf{fma}\left(m \cdot m, 0.25, \ell - \left|n - m\right|\right)}\right)}}} \]

       11. rec-exp 56.2%

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

       12. rem-log-exp 56.2%

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

   12. Simplified 56.2%

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

   if -1.34999999999999994e-179 < n < -9.49999999999999941e-257 or 1.44999999999999994e-126 < n < 52

   1. Initial program 76.7%

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

   3. Simplified 76.7%

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

   4. Taylor expanded in M around inf 67.6%

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

      1. unpow2 67.6%

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

   6. Simplified 67.6%

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

   7. Taylor expanded in K around 0 78.5%

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

      1. cos-neg 24.0%

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

   9. Simplified 78.5%

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

   if -9.49999999999999941e-257 < n < 1.44999999999999994e-126

   1. Initial program 82.2%

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

   3. Simplified 82.2%

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

   4. Taylor expanded in m around inf 71.5%

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

      1. *-commutative 71.5%

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

      2. unpow2 71.5%

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

   6. Simplified 71.5%

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

   7. Taylor expanded in K around 0 85.4%

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

      1. cos-neg 40.6%

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

   9. Simplified 85.4%

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

   if 52 < n

   1. Initial program 68.1%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in n around inf 58.1%

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

      1. *-commutative 58.1%

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

      2. unpow2 58.1%

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

   6. Simplified 58.1%

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

   7. Taylor expanded in K around 0 82.9%

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

      1. cos-neg 82.9%

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

   9. Simplified 82.9%

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

   10. Taylor expanded in n around inf 98.6%

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

       1. unpow2 98.6%

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

   12. Simplified 98.6%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.35 \cdot 10^{-179}:\\ \;\;\;\;e^{\left|n - m\right| - \left(\ell + 0.25 \cdot \left(m \cdot m\right)\right)}\\ \mathbf{elif}\;n \leq -9.5 \cdot 10^{-257}:\\ \;\;\;\;\frac{\cos M}{e^{\left(\ell - \left|n - m\right|\right) + M \cdot M}}\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{-126}:\\ \;\;\;\;\frac{\cos M}{e^{\left(\ell - \left|n - m\right|\right) + 0.25 \cdot \left(m \cdot m\right)}}\\ \mathbf{elif}\;n \leq 52:\\ \;\;\;\;\frac{\cos M}{e^{\left(\ell - \left|n - m\right|\right) + M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \]

Alternative 4: 74.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 0.42:\\ \;\;\;\;e^{\left|n - m\right| - \left(\ell + 0.25 \cdot \left(m \cdot m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 0.42)
   (exp (- (fabs (- n m)) (+ l (* 0.25 (* m m)))))
   (/ (cos M) (exp (* 0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 0.42) {
		tmp = exp((fabs((n - m)) - (l + (0.25 * (m * m)))));
	} else {
		tmp = cos(M) / exp((0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 0.42d0) then
        tmp = exp((abs((n - m)) - (l + (0.25d0 * (m * m)))))
    else
        tmp = cos(m_1) / exp((0.25d0 * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 0.42) {
		tmp = Math.exp((Math.abs((n - m)) - (l + (0.25 * (m * m)))));
	} else {
		tmp = Math.cos(M) / Math.exp((0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 0.42:
		tmp = math.exp((math.fabs((n - m)) - (l + (0.25 * (m * m)))))
	else:
		tmp = math.cos(M) / math.exp((0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 0.42)
		tmp = exp(Float64(abs(Float64(n - m)) - Float64(l + Float64(0.25 * Float64(m * m)))));
	else
		tmp = Float64(cos(M) / exp(Float64(0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 0.42)
		tmp = exp((abs((n - m)) - (l + (0.25 * (m * m)))));
	else
		tmp = cos(M) / exp((0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 0.42], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(l + N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 0.419999999999999984

   1. Initial program 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in m around inf 52.0%

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

      1. *-commutative 52.0%

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

      2. unpow2 52.0%

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

   6. Simplified 52.0%

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

   7. Taylor expanded in K around 0 66.9%

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

      1. cos-neg 51.1%

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

   9. Simplified 66.9%

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

   10. Taylor expanded in M around 0 66.3%

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

       1. sub-neg 66.3%

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

       2. mul-1-neg 66.3%

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

       3. +-commutative 66.3%

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

       4. mul-1-neg 66.3%

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

       5. sub-neg 66.3%

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

       6. associate-+r- 66.3%

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

       7. *-commutative 66.3%

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

       8. fma-def 66.3%

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

       9. unpow2 66.3%

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

       10. rem-log-exp 66.3%

           \[\leadsto \frac{1}{e^{\color{blue}{\log \left(e^{\mathsf{fma}\left(m \cdot m, 0.25, \ell - \left|n - m\right|\right)}\right)}}} \]

       11. rec-exp 66.3%

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

       12. rem-log-exp 66.3%

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

   12. Simplified 66.3%

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

   if 0.419999999999999984 < n

   1. Initial program 68.1%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in n around inf 58.1%

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

      1. *-commutative 58.1%

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

      2. unpow2 58.1%

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

   6. Simplified 58.1%

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

   7. Taylor expanded in K around 0 82.9%

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

      1. cos-neg 82.9%

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

   9. Simplified 82.9%

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

   10. Taylor expanded in n around inf 98.6%

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

       1. unpow2 98.6%

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

   12. Simplified 98.6%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 0.42:\\ \;\;\;\;e^{\left|n - m\right| - \left(\ell + 0.25 \cdot \left(m \cdot m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \]

Alternative 5: 69.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -54 \lor \neg \left(n \leq 0.42\right):\\ \;\;\;\;\frac{\cos M}{e^{0.25 \cdot \left(n \cdot n\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= n -54.0) (not (<= n 0.42)))
   (/ (cos M) (exp (* 0.25 (* n n))))
   (/ (cos M) (exp l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -54.0) || !(n <= 0.42)) {
		tmp = cos(M) / exp((0.25 * (n * n)));
	} else {
		tmp = cos(M) / exp(l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((n <= (-54.0d0)) .or. (.not. (n <= 0.42d0))) then
        tmp = cos(m_1) / exp((0.25d0 * (n * n)))
    else
        tmp = cos(m_1) / exp(l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -54.0) || !(n <= 0.42)) {
		tmp = Math.cos(M) / Math.exp((0.25 * (n * n)));
	} else {
		tmp = Math.cos(M) / Math.exp(l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (n <= -54.0) or not (n <= 0.42):
		tmp = math.cos(M) / math.exp((0.25 * (n * n)))
	else:
		tmp = math.cos(M) / math.exp(l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((n <= -54.0) || !(n <= 0.42))
		tmp = Float64(cos(M) / exp(Float64(0.25 * Float64(n * n))));
	else
		tmp = Float64(cos(M) / exp(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((n <= -54.0) || ~((n <= 0.42)))
		tmp = cos(M) / exp((0.25 * (n * n)));
	else
		tmp = cos(M) / exp(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[n, -54.0], N[Not[LessEqual[n, 0.42]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -54 or 0.419999999999999984 < n

   1. Initial program 67.2%

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

   3. Simplified 67.2%

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

   4. Taylor expanded in n around inf 57.4%

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

      1. *-commutative 57.4%

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

      2. unpow2 57.4%

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

   6. Simplified 57.4%

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

   7. Taylor expanded in K around 0 81.2%

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

      1. cos-neg 81.2%

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

   9. Simplified 81.2%

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

   10. Taylor expanded in n around inf 97.0%

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

       1. unpow2 97.0%

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

   12. Simplified 97.0%

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

   if -54 < n < 0.419999999999999984

   1. Initial program 81.9%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in n around inf 34.8%

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

      1. *-commutative 34.8%

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

      2. unpow2 34.8%

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

   6. Simplified 34.8%

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

   7. Taylor expanded in K around 0 37.0%

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

      1. cos-neg 37.0%

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

   9. Simplified 37.0%

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

   10. Taylor expanded in l around inf 42.7%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -54 \lor \neg \left(n \leq 0.42\right):\\ \;\;\;\;\frac{\cos M}{e^{0.25 \cdot \left(n \cdot n\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \]

Alternative 6: 35.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

3. Simplified 74.3%

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

4. Taylor expanded in n around inf 46.4%

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

   1. *-commutative 46.4%

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

   2. unpow2 46.4%

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

6. Simplified 46.4%

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

7. Taylor expanded in K around 0 59.6%

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

   1. cos-neg 59.6%

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

9. Simplified 59.6%

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

10. Taylor expanded in l around inf 38.6%

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

11. Final simplification 38.6%

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

Reproduce

herbie shell --seed 2023182 
(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)))))))