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

↓

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

Error

The X axis uses an exponential scale — Bits error versus `K`

Derivation

Initial program 14.9
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
Taylor expanded around 0 1.4
\[\leadsto \color{blue}{1} \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
Using strategy rm
Applied add-cbrt-cube1.4
\[\leadsto 1 \cdot e^{\color{blue}{\sqrt[3]{\left(\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)\right) \cdot \left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)\right)\right) \cdot \left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)\right)}}}\]
Applied simplify1.4
\[\leadsto 1 \cdot e^{\sqrt[3]{\color{blue}{{\left(\left(\frac{n + m}{2} - M\right) \cdot \left(-\left(\frac{n + m}{2} - M\right)\right) - \left(\ell - \left|m - n\right|\right)\right)}^{3}}}}\]
Removed slow pow expressions.

Runtime

herbie shell --seed '#(1063185673 2139736501 2393378123 1907444849 1070993796 1007244912)' 
(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)))))))