Average Error: 15.0 → 1.2
Time: 1.2m
Precision: 64
Internal Precision: 128
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
↓
\[\sqrt[3]{\left(e^{\left(\sqrt{\left|m - n\right|} + \left(\frac{n + m}{2} - M\right)\right) \cdot \left(\sqrt{\left|m - n\right|} - \left(\frac{n + m}{2} - M\right)\right) - \ell} \cdot e^{\left(\sqrt{\left|m - n\right|} + \left(\frac{n + m}{2} - M\right)\right) \cdot \left(\sqrt{\left|m - n\right|} - \left(\frac{n + m}{2} - M\right)\right) - \ell}\right) \cdot \sqrt[3]{e^{\left(\sqrt{\left|m - n\right|} + \left(\frac{n + m}{2} - M\right)\right) \cdot \left(\sqrt{\left|m - n\right|} - \left(\frac{n + m}{2} - M\right)\right) - \ell} \cdot \left(e^{\left(\sqrt{\left|m - n\right|} + \left(\frac{n + m}{2} - M\right)\right) \cdot \left(\sqrt{\left|m - n\right|} - \left(\frac{n + m}{2} - M\right)\right) - \ell} \cdot e^{\left(\sqrt{\left|m - n\right|} + \left(\frac{n + m}{2} - M\right)\right) \cdot \left(\sqrt{\left|m - n\right|} - \left(\frac{n + m}{2} - M\right)\right) - \ell}\right)}}\]

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 15.0

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

  2. Simplified15.0

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

  3. Taylor expanded around 0 1.2

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

  5. Applied add-sqr-sqrt1.2

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

  6. Applied difference-of-squares1.2

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

  8. Applied add-cbrt-cube1.2

    \[\leadsto \color{blue}{\sqrt[3]{\left(e^{\left(\sqrt{\left|m - n\right|} + \left(\frac{m + n}{2} - M\right)\right) \cdot \left(\sqrt{\left|m - n\right|} - \left(\frac{m + n}{2} - M\right)\right) - \ell} \cdot e^{\left(\sqrt{\left|m - n\right|} + \left(\frac{m + n}{2} - M\right)\right) \cdot \left(\sqrt{\left|m - n\right|} - \left(\frac{m + n}{2} - M\right)\right) - \ell}\right) \cdot e^{\left(\sqrt{\left|m - n\right|} + \left(\frac{m + n}{2} - M\right)\right) \cdot \left(\sqrt{\left|m - n\right|} - \left(\frac{m + n}{2} - M\right)\right) - \ell}}} \cdot 1\]
  9. Using strategy rm

  10. Applied add-cbrt-cube1.2

    \[\leadsto \sqrt[3]{\left(e^{\left(\sqrt{\left|m - n\right|} + \left(\frac{m + n}{2} - M\right)\right) \cdot \left(\sqrt{\left|m - n\right|} - \left(\frac{m + n}{2} - M\right)\right) - \ell} \cdot e^{\left(\sqrt{\left|m - n\right|} + \left(\frac{m + n}{2} - M\right)\right) \cdot \left(\sqrt{\left|m - n\right|} - \left(\frac{m + n}{2} - M\right)\right) - \ell}\right) \cdot \color{blue}{\sqrt[3]{\left(e^{\left(\sqrt{\left|m - n\right|} + \left(\frac{m + n}{2} - M\right)\right) \cdot \left(\sqrt{\left|m - n\right|} - \left(\frac{m + n}{2} - M\right)\right) - \ell} \cdot e^{\left(\sqrt{\left|m - n\right|} + \left(\frac{m + n}{2} - M\right)\right) \cdot \left(\sqrt{\left|m - n\right|} - \left(\frac{m + n}{2} - M\right)\right) - \ell}\right) \cdot e^{\left(\sqrt{\left|m - n\right|} + \left(\frac{m + n}{2} - M\right)\right) \cdot \left(\sqrt{\left|m - n\right|} - \left(\frac{m + n}{2} - M\right)\right) - \ell}}}} \cdot 1\]

  11. Final simplification1.2

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

Reproduce

herbie shell --seed 2019089 
(FPCore (K m n M l)
  :name "Maksimov and Kolovsky, Equation (32)"
  (* (cos (- (/ (* K (+ m n)) 2) M)) (exp (- (- (pow (- (/ (+ m n) 2) M) 2)) (- l (fabs (- m n)))))))