Average Error: 15.4 → 1.3
Time: 18.6s
Precision: binary64
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
↓
\[\cos M \cdot e^{\left(-{\left({\left(\sqrt[3]{\frac{m + n}{2} - M}\right)}^{3}\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
(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
 (*
  (cos M)
  (exp
   (-
    (- (pow (pow (cbrt (- (/ (+ m n) 2.0) M)) 3.0) 2.0))
    (- l (fabs (- m n)))))))
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 cos(M) * exp((-pow(pow(cbrt((((m + n) / 2.0) - M)), 3.0), 2.0) - (l - fabs((m - n)))));
}

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.cos(M) * Math.exp((-Math.pow(Math.pow(Math.cbrt((((m + n) / 2.0) - M)), 3.0), 2.0) - (l - Math.abs((m - n)))));
}

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 Float64(cos(M) * exp(Float64(Float64(-((cbrt(Float64(Float64(Float64(m + n) / 2.0) - M)) ^ 3.0) ^ 2.0)) - Float64(l - abs(Float64(m - n))))))
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[(N[Cos[M], $MachinePrecision] * N[Exp[N[((-N[Power[N[Power[N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], 2.0], $MachinePrecision]) - N[(l - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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)}
↓
\cos M \cdot e^{\left(-{\left({\left(\sqrt[3]{\frac{m + n}{2} - M}\right)}^{3}\right)}^{2}\right) - \left(\ell - \left|m - n\right|\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 15.4

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

  2. Taylor expanded in K around 0 1.3

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

  3. Simplified1.3

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

  4. Applied egg-rr1.3

    \[\leadsto \cos M \cdot e^{\left(-{\color{blue}{\left({\left(\sqrt[3]{\frac{m + n}{2} - M}\right)}^{3}\right)}}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

  5. Final simplification1.3

    \[\leadsto \cos M \cdot e^{\left(-{\left({\left(\sqrt[3]{\frac{m + n}{2} - M}\right)}^{3}\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

Reproduce

herbie shell --seed 2022152 
(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)))))))