Average Error: 15.0 → 1.6
Time: 6.8s
Precision: binary64
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
↓
\[e^{\left|n - m\right| - \left({\left(\frac{n + m}{2} - M\right)}^{2} + \ell\right)} \]
(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)))))))
↓
(FPCore (K m n M l)
 :precision binary64
 (exp (- (fabs (- n m)) (+ (pow (- (/ (+ n m) 2.0) M) 2.0) l))))
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)))));
}
↓
double code(double K, double m, double n, double M, double l) {
	return exp((fabs((n - m)) - (pow((((n + m) / 2.0) - M), 2.0) + l)));
}

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

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

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

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

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

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

\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)}
↓
e^{\left|n - m\right| - \left({\left(\frac{n + m}{2} - M\right)}^{2} + \ell\right)}

Error

Bits error versus K

Bits error versus m

Bits error versus n

Bits error versus M

Bits error versus l

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.0

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

  2. Simplified15.0

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

  3. Taylor expanded in n around inf 9.7

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

  4. Taylor expanded in n around 0 1.6

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

  5. Final simplification1.6

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

Reproduce

herbie shell --seed 2022170 
(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)))))))