Average Error: 14.9 → 1.3
Time: 10.7s
Precision: binary64
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
↓
\[{\left(\sqrt[3]{\cos M}\right)}^{2} \cdot \left(\sqrt[3]{\cos M} \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{m + n}{2} - M\right)}^{2}\right)}\right)\]
\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)}
↓
{\left(\sqrt[3]{\cos M}\right)}^{2} \cdot \left(\sqrt[3]{\cos M} \cdot e^{\left|m - n\right| - \left(\ell + {\left(\frac{m + n}{2} - M\right)}^{2}\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
 (*
  (pow (cbrt (cos M)) 2.0)
  (*
   (cbrt (cos M))
   (exp (- (fabs (- m n)) (+ l (pow (- (/ (+ m n) 2.0) M) 2.0)))))))
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 pow(cbrt(cos(M)), 2.0) * (cbrt(cos(M)) * exp(fabs(m - n) - (l + pow((((m + n) / 2.0) - M), 2.0))));
}

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 14.9

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

  2. Simplified14.9

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

  3. Taylor expanded around 0 1.3

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

  4. Simplified1.3

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

  6. Applied add-cube-cbrt_binary64_7951.3

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

  7. Applied associate-*l*_binary64_7011.3

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

  8. Simplified1.3

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

  10. Applied pow2_binary64_8411.3

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

  11. Final simplification1.3

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

Reproduce

herbie shell --seed 2021024 
(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)))))))