Average Error: 15.0 → 1.2
Time: 5.6m
Precision: 64
Internal Precision: 1664
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
↓
\[1 \cdot \left(\left(\sqrt[3]{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}} \cdot \sqrt[3]{\left(\sqrt[3]{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}} \cdot \sqrt[3]{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}}\right) \cdot \sqrt[3]{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}}}\right) \cdot \sqrt[3]{e^{\left(-(e^{\log_* (1 + {\left(\frac{m + n}{2} - M\right)}^{2})} - 1)^*\right) - \left(\ell - \left|m - n\right|\right)}}\right)\]

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 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. Taylor expanded around 0 1.2

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

  4. Applied add-cube-cbrt1.2

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

  6. Applied expm1-log1p-u1.2

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

  8. Applied add-cube-cbrt1.2

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

Runtime

Time bar (total: 5.6m)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' +o rules:numerics
(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)))))))