Average Error: 15.4 → 1.4
Time: 18.2s
Precision: binary64
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\cos M \cdot e^{\left|m - n\right| - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \ell\right)}\right)\right) \]
\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)}
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\cos M \cdot e^{\left|m - n\right| - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \ell\right)}\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
 (expm1
  (log1p
   (*
    (cos M)
    (exp (- (fabs (- m n)) (+ (pow (- (/ (+ m n) 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 expm1(log1p((cos(M) * exp((fabs((m - n)) - (pow((((m + n) / 2.0) - M), 2.0) + l))))));
}

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. Simplified15.4

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

  3. Taylor expanded in K around 0 1.4

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

  4. Simplified1.4

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

  5. Applied expm1-log1p-u_binary641.4

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos M \cdot e^{\left|m - n\right| - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \ell\right)}\right)\right)} \]

  6. Final simplification1.4

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos M \cdot e^{\left|m - n\right| - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \ell\right)}\right)\right) \]

Reproduce

herbie shell --seed 2022160 
(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)))))))