Average Error: 15.5 → 1.4
Time: 41.2s
Precision: 64
Internal Precision: 1600
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
↓
\[e^{\left(-\left(\log \left(\sqrt[3]{e^{{\left(\frac{m + n}{2} - M\right)}^{2}}} \cdot \sqrt[3]{e^{{\left(\frac{m + n}{2} - M\right)}^{2}}}\right) + \log \left(\sqrt[3]{e^{{\left(\frac{m + n}{2} - M\right)}^{2}}}\right)\right)\right) - \left(\ell - \left|m - n\right|\right)}\]

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 15.5

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

  2. Taylor expanded around 0 1.4

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

  4. Applied add-log-exp1.4

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

  6. Applied add-cube-cbrt1.4

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

  7. Applied log-prod1.4

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

  8. Final simplification1.4

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

Runtime

Time bar (total: 41.2s)Debug logProfile

herbie shell --seed 2018249 
(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)))))))