Average Error: 15.1 → 1.5
Time: 33.1s
Precision: binary64
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
↓
\[\begin{array}{l} t_0 := \ell + {\left(\frac{m + n}{2} - M\right)}^{2}\\ \cos M \cdot e^{\left|m - n\right| - \left(2 \cdot \left(0.3333333333333333 \cdot t_0\right) + \log \left(\sqrt[3]{e^{t_0}}\right)\right)} \end{array} \]
\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)}
↓
\begin{array}{l}
t_0 := \ell + {\left(\frac{m + n}{2} - M\right)}^{2}\\
\cos M \cdot e^{\left|m - n\right| - \left(2 \cdot \left(0.3333333333333333 \cdot t_0\right) + \log \left(\sqrt[3]{e^{t_0}}\right)\right)}
\end{array}

(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
 (let* ((t_0 (+ l (pow (- (/ (+ m n) 2.0) M) 2.0))))
   (*
    (cos M)
    (exp
     (-
      (fabs (- m n))
      (+ (* 2.0 (* 0.3333333333333333 t_0)) (log (cbrt (exp t_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) {
	double t_0 = l + pow((((m + n) / 2.0) - M), 2.0);
	return cos(M) * exp(fabs(m - n) - ((2.0 * (0.3333333333333333 * t_0)) + log(cbrt(exp(t_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 1.5

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

  4. Simplified1.5

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

  5. Applied add-log-exp_binary641.5

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

  6. Applied add-cube-cbrt_binary641.5

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

  7. Applied log-prod_binary641.5

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

  8. Simplified1.5

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

  9. Simplified1.5

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

  10. Applied pow1/3_binary641.5

    \[\leadsto \cos M \cdot e^{\left|m - n\right| - \left(2 \cdot \log \color{blue}{\left({\left(e^{\ell + {\left(\frac{n + m}{2} - M\right)}^{2}}\right)}^{0.3333333333333333}\right)} + \log \left(\sqrt[3]{e^{\ell + {\left(\frac{n + m}{2} - M\right)}^{2}}}\right)\right)} \]

  11. Applied log-pow_binary641.5

    \[\leadsto \cos M \cdot e^{\left|m - n\right| - \left(2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(e^{\ell + {\left(\frac{n + m}{2} - M\right)}^{2}}\right)\right)} + \log \left(\sqrt[3]{e^{\ell + {\left(\frac{n + m}{2} - M\right)}^{2}}}\right)\right)} \]

  12. Simplified1.5

    \[\leadsto \cos M \cdot e^{\left|m - n\right| - \left(2 \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(\ell + {\left(\frac{m + n}{2} - M\right)}^{2}\right)}\right) + \log \left(\sqrt[3]{e^{\ell + {\left(\frac{n + m}{2} - M\right)}^{2}}}\right)\right)} \]

  13. Final simplification1.5

    \[\leadsto \cos M \cdot e^{\left|m - n\right| - \left(2 \cdot \left(0.3333333333333333 \cdot \left(\ell + {\left(\frac{m + n}{2} - M\right)}^{2}\right)\right) + \log \left(\sqrt[3]{e^{\ell + {\left(\frac{m + n}{2} - M\right)}^{2}}}\right)\right)} \]

Reproduce

herbie shell --seed 2022087 
(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)))))))