Migdal et al, Equation (51)

Average Error: 0.4 → 0.5

Time: 30.8s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

↓

\[\left(\left(1 \cdot \sqrt[3]{\frac{1}{k}}\right) \cdot \sqrt{\frac{1}{\left|\sqrt[3]{k}\right|}}\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

C

double f(double k, double n) {
        double r174508 = 1.0;
        double r174509 = k;
        double r174510 = sqrt(r174509);
        double r174511 = r174508 / r174510;
        double r174512 = 2.0;
        double r174513 = atan2(1.0, 0.0);
        double r174514 = r174512 * r174513;
        double r174515 = n;
        double r174516 = r174514 * r174515;
        double r174517 = r174508 - r174509;
        double r174518 = r174517 / r174512;
        double r174519 = pow(r174516, r174518);
        double r174520 = r174511 * r174519;
        return r174520;
}

↓

double f(double k, double n) {
        double r174521 = 1.0;
        double r174522 = 1.0;
        double r174523 = k;
        double r174524 = r174522 / r174523;
        double r174525 = cbrt(r174524);
        double r174526 = r174521 * r174525;
        double r174527 = cbrt(r174523);
        double r174528 = fabs(r174527);
        double r174529 = r174522 / r174528;
        double r174530 = sqrt(r174529);
        double r174531 = r174526 * r174530;
        double r174532 = 2.0;
        double r174533 = atan2(1.0, 0.0);
        double r174534 = r174532 * r174533;
        double r174535 = n;
        double r174536 = r174534 * r174535;
        double r174537 = r174521 - r174523;
        double r174538 = r174537 / r174532;
        double r174539 = pow(r174536, r174538);
        double r174540 = r174531 * r174539;
        return r174540;
}

Error

Bits error versus k

Bits error versus n

Derivation

1. Initial program 0.4

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
2. Using strategy rm

3. Applied add-sqr-sqrt 0.4

   \[\leadsto \frac{1}{\sqrt{\color{blue}{\sqrt{k} \cdot \sqrt{k}}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

4. Applied sqrt-prod 0.5

   \[\leadsto \frac{1}{\color{blue}{\sqrt{\sqrt{k}} \cdot \sqrt{\sqrt{k}}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

5. Applied *-un-lft-identity 0.5

   \[\leadsto \frac{\color{blue}{1 \cdot 1}}{\sqrt{\sqrt{k}} \cdot \sqrt{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

6. Applied times-frac 0.5

   \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\sqrt{k}}} \cdot \frac{1}{\sqrt{\sqrt{k}}}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
7. Using strategy rm

8. Applied add-cube-cbrt 0.5

   \[\leadsto \left(\frac{1}{\sqrt{\sqrt{k}}} \cdot \frac{1}{\sqrt{\sqrt{\color{blue}{\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right) \cdot \sqrt[3]{k}}}}}\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

9. Applied sqrt-prod 0.5

   \[\leadsto \left(\frac{1}{\sqrt{\sqrt{k}}} \cdot \frac{1}{\sqrt{\color{blue}{\sqrt{\sqrt[3]{k} \cdot \sqrt[3]{k}} \cdot \sqrt{\sqrt[3]{k}}}}}\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

10. Applied sqrt-prod 0.5

    \[\leadsto \left(\frac{1}{\sqrt{\sqrt{k}}} \cdot \frac{1}{\color{blue}{\sqrt{\sqrt{\sqrt[3]{k} \cdot \sqrt[3]{k}}} \cdot \sqrt{\sqrt{\sqrt[3]{k}}}}}\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

11. Applied add-cube-cbrt 0.5

    \[\leadsto \left(\frac{1}{\sqrt{\sqrt{k}}} \cdot \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\sqrt{\sqrt{\sqrt[3]{k} \cdot \sqrt[3]{k}}} \cdot \sqrt{\sqrt{\sqrt[3]{k}}}}\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

12. Applied times-frac 0.5

    \[\leadsto \left(\frac{1}{\sqrt{\sqrt{k}}} \cdot \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{\sqrt{\sqrt[3]{k} \cdot \sqrt[3]{k}}}} \cdot \frac{\sqrt[3]{1}}{\sqrt{\sqrt{\sqrt[3]{k}}}}\right)}\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

13. Applied associate-*r* 0.5

    \[\leadsto \color{blue}{\left(\left(\frac{1}{\sqrt{\sqrt{k}}} \cdot \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{\sqrt{\sqrt[3]{k} \cdot \sqrt[3]{k}}}}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt{\sqrt{\sqrt[3]{k}}}}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

14. Simplified 0.5

    \[\leadsto \left(\color{blue}{\frac{\frac{\sqrt[3]{1}}{\frac{\sqrt{\left|\sqrt[3]{k}\right|}}{\sqrt[3]{1}}}}{\sqrt{\sqrt{k}}}} \cdot \frac{\sqrt[3]{1}}{\sqrt{\sqrt{\sqrt[3]{k}}}}\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

15. Taylor expanded around 0 2.9

    \[\leadsto \color{blue}{\left(1 \cdot \left(\sqrt{\frac{1}{\left|{k}^{\frac{1}{3}}\right|}} \cdot {\left(\frac{1}{k}\right)}^{\frac{1}{3}}\right)\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

16. Simplified 0.5

    \[\leadsto \color{blue}{\left(\left(1 \cdot \sqrt[3]{\frac{1}{k}}\right) \cdot \sqrt{\frac{1}{\left|\sqrt[3]{k}\right|}}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

17. Final simplification 0.5

    \[\leadsto \left(\left(1 \cdot \sqrt[3]{\frac{1}{k}}\right) \cdot \sqrt{\frac{1}{\left|\sqrt[3]{k}\right|}}\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

Reproduce

herbie shell --seed 2019351 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))