Toniolo and Linder, Equation (10+)

Average Error: 32.1 → 16.3

Time: 23.0s

Precision: 64

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

Math

\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

↓

\[\frac{2}{\left(\tan k \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}} \cdot \sqrt[3]{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}}}\right) \cdot \left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}}} \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \ell\]

C

double f(double t, double l, double k) {
        double r73647 = 2.0;
        double r73648 = t;
        double r73649 = 3.0;
        double r73650 = pow(r73648, r73649);
        double r73651 = l;
        double r73652 = r73651 * r73651;
        double r73653 = r73650 / r73652;
        double r73654 = k;
        double r73655 = sin(r73654);
        double r73656 = r73653 * r73655;
        double r73657 = tan(r73654);
        double r73658 = r73656 * r73657;
        double r73659 = 1.0;
        double r73660 = r73654 / r73648;
        double r73661 = pow(r73660, r73647);
        double r73662 = r73659 + r73661;
        double r73663 = r73662 + r73659;
        double r73664 = r73658 * r73663;
        double r73665 = r73647 / r73664;
        return r73665;
}

↓

double f(double t, double l, double k) {
        double r73666 = 2.0;
        double r73667 = k;
        double r73668 = tan(r73667);
        double r73669 = t;
        double r73670 = cbrt(r73669);
        double r73671 = r73670 * r73670;
        double r73672 = 3.0;
        double r73673 = 2.0;
        double r73674 = r73672 / r73673;
        double r73675 = pow(r73671, r73674);
        double r73676 = l;
        double r73677 = pow(r73670, r73672);
        double r73678 = sin(r73667);
        double r73679 = r73677 * r73678;
        double r73680 = r73676 / r73679;
        double r73681 = cbrt(r73680);
        double r73682 = r73681 * r73681;
        double r73683 = r73675 / r73682;
        double r73684 = r73668 * r73683;
        double r73685 = r73675 / r73681;
        double r73686 = 1.0;
        double r73687 = r73667 / r73669;
        double r73688 = pow(r73687, r73666);
        double r73689 = fma(r73673, r73686, r73688);
        double r73690 = r73685 * r73689;
        double r73691 = r73684 * r73690;
        double r73692 = r73666 / r73691;
        double r73693 = r73692 * r73676;
        return r73693;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Initial program 32.1

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

2. Simplified 27.6

   \[\leadsto \color{blue}{\frac{2}{\left(\tan k \cdot \frac{{t}^{3} \cdot \sin k}{\ell}\right) \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \ell}\]
3. Using strategy rm

4. Applied add-cube-cbrt 27.9

   \[\leadsto \frac{2}{\left(\tan k \cdot \frac{{\color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}}^{3} \cdot \sin k}{\ell}\right) \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \ell\]

5. Applied unpow-prod-down 27.9

   \[\leadsto \frac{2}{\left(\tan k \cdot \frac{\color{blue}{\left({\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)} \cdot \sin k}{\ell}\right) \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \ell\]

6. Applied associate-*l* 26.8

   \[\leadsto \frac{2}{\left(\tan k \cdot \frac{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot \left({\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k\right)}}{\ell}\right) \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \ell\]
7. Using strategy rm

8. Applied associate-/l* 23.9

   \[\leadsto \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}}}\right) \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \ell\]
9. Using strategy rm

10. Applied add-cube-cbrt 23.9

    \[\leadsto \frac{2}{\left(\tan k \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\color{blue}{\left(\sqrt[3]{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}} \cdot \sqrt[3]{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}}\right) \cdot \sqrt[3]{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}}}}\right) \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \ell\]

11. Applied sqr-pow 23.9

    \[\leadsto \frac{2}{\left(\tan k \cdot \frac{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}}{\left(\sqrt[3]{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}} \cdot \sqrt[3]{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}}\right) \cdot \sqrt[3]{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}}}\right) \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \ell\]

12. Applied times-frac 21.7

    \[\leadsto \frac{2}{\left(\tan k \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}} \cdot \sqrt[3]{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}}} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}}}\right)}\right) \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \ell\]

13. Applied associate-*r* 19.5

    \[\leadsto \frac{2}{\color{blue}{\left(\left(\tan k \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}} \cdot \sqrt[3]{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}}}\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}}}\right)} \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \ell\]
14. Using strategy rm

15. Applied associate-*l* 16.3

    \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}} \cdot \sqrt[3]{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}}}\right) \cdot \left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}}} \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \cdot \ell\]

16. Final simplification 16.3

    \[\leadsto \frac{2}{\left(\tan k \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}} \cdot \sqrt[3]{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}}}\right) \cdot \left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}}} \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \ell\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))