Maksimov and Kolovsky, Equation (32)

Average Error: 16.0 → 1.5

Time: 5.2s

Precision: binary64

Math

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

↓

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

Error

Bits error versus K

Bits error versus m

Bits error versus n

Bits error versus M

Bits error versus l

Derivation

1. Initial program 16.0

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

2. Simplified 16.0

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

3. Taylor expanded around 0 1.5

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

4. Final simplification 1.5

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

Reproduce

herbie shell --seed 2020182 
(FPCore (K m n M l)
  :name "Maksimov and Kolovsky, Equation (32)"
  :precision binary64
  (* (cos (- (/ (* K (+ m n)) 2.0) M)) (exp (- (neg (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))