Average Error: 15.1 → 1.4
Time: 9.0s
Precision: binary64
\[[m, n]=\mathsf{sort}([m, n])\]
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
↓
\[\cos M \cdot e^{\left(m - n\right) - \left(\ell + {\left(\frac{m + n}{2} - M\right)}^{2}\right)} \]
\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)}
↓
\cos M \cdot e^{\left(m - n\right) - \left(\ell + {\left(\frac{m + n}{2} - M\right)}^{2}\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 (- (- 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 cos(M) * exp((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 15.1

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

  2. Simplified15.1

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

  3. Taylor expanded in K around 0 1.3

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

  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. Applied add-cube-cbrt_binary641.3

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

  6. Applied add-cube-cbrt_binary641.3

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

  7. Applied prod-diff_binary6440.4

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

  8. Simplified40.4

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

  9. Simplified1.4

    \[\leadsto \cos M \cdot e^{\left(\left(m - n\right) - \left(\ell + {\left(\frac{n + m}{2} - M\right)}^{2}\right)\right) + \color{blue}{0}} \]

  10. Final simplification1.4

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

Reproduce

herbie shell --seed 2021344 
(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)))))))