Maksimov and Kolovsky, Equation (32)

Average Error: 14.7 → 1.2

Time: 43.2s

Precision: 64

Math

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

↓

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

C

double f(double K, double m, double n, double M, double l) {
        double r71612 = K;
        double r71613 = m;
        double r71614 = n;
        double r71615 = r71613 + r71614;
        double r71616 = r71612 * r71615;
        double r71617 = 2.0;
        double r71618 = r71616 / r71617;
        double r71619 = M;
        double r71620 = r71618 - r71619;
        double r71621 = cos(r71620);
        double r71622 = r71615 / r71617;
        double r71623 = r71622 - r71619;
        double r71624 = pow(r71623, r71617);
        double r71625 = -r71624;
        double r71626 = l;
        double r71627 = r71613 - r71614;
        double r71628 = fabs(r71627);
        double r71629 = r71626 - r71628;
        double r71630 = r71625 - r71629;
        double r71631 = exp(r71630);
        double r71632 = r71621 * r71631;
        return r71632;
}

↓

double f(double __attribute__((unused)) K, double m, double n, double M, double l) {
        double r71633 = m;
        double r71634 = n;
        double r71635 = r71633 + r71634;
        double r71636 = 2.0;
        double r71637 = r71635 / r71636;
        double r71638 = M;
        double r71639 = r71637 - r71638;
        double r71640 = pow(r71639, r71636);
        double r71641 = -r71640;
        double r71642 = l;
        double r71643 = r71633 - r71634;
        double r71644 = fabs(r71643);
        double r71645 = r71642 - r71644;
        double r71646 = r71641 - r71645;
        double r71647 = exp(r71646);
        double r71648 = cbrt(r71647);
        double r71649 = r71640 + r71642;
        double r71650 = r71644 - r71649;
        double r71651 = exp(r71650);
        double r71652 = cbrt(r71651);
        double r71653 = r71652 * r71652;
        double r71654 = r71653 * r71652;
        double r71655 = cbrt(r71654);
        double r71656 = r71648 * r71655;
        double r71657 = cbrt(r71653);
        double r71658 = cbrt(r71652);
        double r71659 = r71657 * r71658;
        double r71660 = r71656 * r71659;
        return r71660;
}

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 14.7

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

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-cbrt 1.2

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

6. Applied add-cube-cbrt 1.2

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

7. Applied cbrt-prod 1.2

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

8. Simplified 1.2

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

9. Simplified 1.2

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

11. Applied add-cube-cbrt 1.2

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

12. Simplified 1.2

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

13. Simplified 1.2

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

14. Final simplification 1.2

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

Reproduce

herbie shell --seed 2019195 
(FPCore (K m n M l)
  :name "Maksimov and Kolovsky, Equation (32)"
  (* (cos (- (/ (* K (+ m n)) 2.0) M)) (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))