Average Error: 14.9 → 1.3
Time: 1.4m
Precision: 64
Internal Precision: 1600
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
↓
\[1 \cdot e^{\left(-\left(\left(\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \left(\sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}} \cdot \sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}} \cdot \sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}}\right) \cdot \sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}}\right)\right) - \left(\ell - \left|m - n\right|\right)}\]

Error

Bits error versus K

Bits error versus m

Bits error versus n

Bits error versus M

Bits error versus l

Derivation

  1. Initial program 14.9

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

  2. Taylor expanded around 0 1.3

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

  4. Applied add-cube-cbrt1.3

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

  6. Applied add-cube-cbrt1.3

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

  7. Applied associate-*r*1.3

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

  9. Applied add-cube-cbrt1.3

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed '#(1072330854 3074818769 591214268 3603999196 3863745332 3332387116)' 
(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)))))))