Toniolo and Linder, Equation (10-)

Average Error: 47.4 → 7.7

Time: 3.5m

Precision: 64

Math

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

↓

\[\frac{\frac{1}{\frac{t}{\ell}}}{k} \cdot \left(\left(\frac{\frac{\frac{1}{\frac{\sqrt[3]{t}}{\ell}}}{\sin k}}{\frac{1}{\sqrt[3]{t}}} \cdot \frac{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right) \cdot \left(\frac{\cos k}{k \cdot \sin k} \cdot 2\right)\right)\]

C

double f(double t, double l, double k) {
        double r7461489 = 2.0;
        double r7461490 = t;
        double r7461491 = 3.0;
        double r7461492 = pow(r7461490, r7461491);
        double r7461493 = l;
        double r7461494 = r7461493 * r7461493;
        double r7461495 = r7461492 / r7461494;
        double r7461496 = k;
        double r7461497 = sin(r7461496);
        double r7461498 = r7461495 * r7461497;
        double r7461499 = tan(r7461496);
        double r7461500 = r7461498 * r7461499;
        double r7461501 = 1.0;
        double r7461502 = r7461496 / r7461490;
        double r7461503 = pow(r7461502, r7461489);
        double r7461504 = r7461501 + r7461503;
        double r7461505 = r7461504 - r7461501;
        double r7461506 = r7461500 * r7461505;
        double r7461507 = r7461489 / r7461506;
        return r7461507;
}

↓

double f(double t, double l, double k) {
        double r7461508 = 1.0;
        double r7461509 = t;
        double r7461510 = l;
        double r7461511 = r7461509 / r7461510;
        double r7461512 = r7461508 / r7461511;
        double r7461513 = k;
        double r7461514 = r7461512 / r7461513;
        double r7461515 = cbrt(r7461509);
        double r7461516 = r7461515 / r7461510;
        double r7461517 = r7461508 / r7461516;
        double r7461518 = sin(r7461513);
        double r7461519 = r7461517 / r7461518;
        double r7461520 = r7461508 / r7461515;
        double r7461521 = r7461519 / r7461520;
        double r7461522 = r7461515 * r7461515;
        double r7461523 = r7461508 / r7461522;
        double r7461524 = r7461523 / r7461523;
        double r7461525 = r7461521 * r7461524;
        double r7461526 = cos(r7461513);
        double r7461527 = r7461513 * r7461518;
        double r7461528 = r7461526 / r7461527;
        double r7461529 = 2.0;
        double r7461530 = r7461528 * r7461529;
        double r7461531 = r7461525 * r7461530;
        double r7461532 = r7461514 * r7461531;
        return r7461532;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Initial program 47.4

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

2. Simplified 30.8

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

4. Applied *-un-lft-identity 30.8

   \[\leadsto \frac{\frac{\frac{\color{blue}{1 \cdot 2}}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot t}}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t}}\]

5. Applied times-frac 30.8

   \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\frac{t}{\ell} \cdot \frac{t}{\ell}} \cdot \frac{2}{t}}}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t}}\]

6. Applied times-frac 30.7

   \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\frac{t}{\ell} \cdot \frac{t}{\ell}}}{\sin k} \cdot \frac{\frac{2}{t}}{\tan k}}}{\frac{k}{t} \cdot \frac{k}{t}}\]

7. Applied times-frac 20.0

   \[\leadsto \color{blue}{\frac{\frac{\frac{1}{\frac{t}{\ell} \cdot \frac{t}{\ell}}}{\sin k}}{\frac{k}{t}} \cdot \frac{\frac{\frac{2}{t}}{\tan k}}{\frac{k}{t}}}\]
8. Using strategy rm

9. Applied div-inv 20.0

   \[\leadsto \frac{\frac{\frac{1}{\frac{t}{\ell} \cdot \frac{t}{\ell}}}{\sin k}}{\color{blue}{k \cdot \frac{1}{t}}} \cdot \frac{\frac{\frac{2}{t}}{\tan k}}{\frac{k}{t}}\]

10. Applied *-un-lft-identity 20.0

    \[\leadsto \frac{\frac{\frac{1}{\frac{t}{\ell} \cdot \frac{t}{\ell}}}{\color{blue}{1 \cdot \sin k}}}{k \cdot \frac{1}{t}} \cdot \frac{\frac{\frac{2}{t}}{\tan k}}{\frac{k}{t}}\]

11. Applied *-un-lft-identity 20.0

    \[\leadsto \frac{\frac{\frac{\color{blue}{1 \cdot 1}}{\frac{t}{\ell} \cdot \frac{t}{\ell}}}{1 \cdot \sin k}}{k \cdot \frac{1}{t}} \cdot \frac{\frac{\frac{2}{t}}{\tan k}}{\frac{k}{t}}\]

12. Applied times-frac 19.9

    \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\frac{t}{\ell}} \cdot \frac{1}{\frac{t}{\ell}}}}{1 \cdot \sin k}}{k \cdot \frac{1}{t}} \cdot \frac{\frac{\frac{2}{t}}{\tan k}}{\frac{k}{t}}\]

13. Applied times-frac 19.1

    \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\frac{t}{\ell}}}{1} \cdot \frac{\frac{1}{\frac{t}{\ell}}}{\sin k}}}{k \cdot \frac{1}{t}} \cdot \frac{\frac{\frac{2}{t}}{\tan k}}{\frac{k}{t}}\]

14. Applied times-frac 13.4

    \[\leadsto \color{blue}{\left(\frac{\frac{\frac{1}{\frac{t}{\ell}}}{1}}{k} \cdot \frac{\frac{\frac{1}{\frac{t}{\ell}}}{\sin k}}{\frac{1}{t}}\right)} \cdot \frac{\frac{\frac{2}{t}}{\tan k}}{\frac{k}{t}}\]

15. Applied associate-*l* 12.1

    \[\leadsto \color{blue}{\frac{\frac{\frac{1}{\frac{t}{\ell}}}{1}}{k} \cdot \left(\frac{\frac{\frac{1}{\frac{t}{\ell}}}{\sin k}}{\frac{1}{t}} \cdot \frac{\frac{\frac{2}{t}}{\tan k}}{\frac{k}{t}}\right)}\]

16. Taylor expanded around inf 7.7

    \[\leadsto \frac{\frac{\frac{1}{\frac{t}{\ell}}}{1}}{k} \cdot \left(\frac{\frac{\frac{1}{\frac{t}{\ell}}}{\sin k}}{\frac{1}{t}} \cdot \color{blue}{\left(2 \cdot \frac{\cos k}{k \cdot \sin k}\right)}\right)\]
17. Using strategy rm

18. Applied add-cube-cbrt 8.0

    \[\leadsto \frac{\frac{\frac{1}{\frac{t}{\ell}}}{1}}{k} \cdot \left(\frac{\frac{\frac{1}{\frac{t}{\ell}}}{\sin k}}{\frac{1}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}} \cdot \left(2 \cdot \frac{\cos k}{k \cdot \sin k}\right)\right)\]

19. Applied *-un-lft-identity 8.0

    \[\leadsto \frac{\frac{\frac{1}{\frac{t}{\ell}}}{1}}{k} \cdot \left(\frac{\frac{\frac{1}{\frac{t}{\ell}}}{\sin k}}{\frac{\color{blue}{1 \cdot 1}}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}} \cdot \left(2 \cdot \frac{\cos k}{k \cdot \sin k}\right)\right)\]

20. Applied times-frac 8.0

    \[\leadsto \frac{\frac{\frac{1}{\frac{t}{\ell}}}{1}}{k} \cdot \left(\frac{\frac{\frac{1}{\frac{t}{\ell}}}{\sin k}}{\color{blue}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{1}{\sqrt[3]{t}}}} \cdot \left(2 \cdot \frac{\cos k}{k \cdot \sin k}\right)\right)\]

21. Applied *-un-lft-identity 8.0

    \[\leadsto \frac{\frac{\frac{1}{\frac{t}{\ell}}}{1}}{k} \cdot \left(\frac{\frac{\frac{1}{\frac{t}{\ell}}}{\color{blue}{1 \cdot \sin k}}}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{1}{\sqrt[3]{t}}} \cdot \left(2 \cdot \frac{\cos k}{k \cdot \sin k}\right)\right)\]

22. Applied *-un-lft-identity 8.0

    \[\leadsto \frac{\frac{\frac{1}{\frac{t}{\ell}}}{1}}{k} \cdot \left(\frac{\frac{\frac{1}{\frac{t}{\color{blue}{1 \cdot \ell}}}}{1 \cdot \sin k}}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{1}{\sqrt[3]{t}}} \cdot \left(2 \cdot \frac{\cos k}{k \cdot \sin k}\right)\right)\]

23. Applied add-cube-cbrt 7.7

    \[\leadsto \frac{\frac{\frac{1}{\frac{t}{\ell}}}{1}}{k} \cdot \left(\frac{\frac{\frac{1}{\frac{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}{1 \cdot \ell}}}{1 \cdot \sin k}}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{1}{\sqrt[3]{t}}} \cdot \left(2 \cdot \frac{\cos k}{k \cdot \sin k}\right)\right)\]

24. Applied times-frac 7.7

    \[\leadsto \frac{\frac{\frac{1}{\frac{t}{\ell}}}{1}}{k} \cdot \left(\frac{\frac{\frac{1}{\color{blue}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{1} \cdot \frac{\sqrt[3]{t}}{\ell}}}}{1 \cdot \sin k}}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{1}{\sqrt[3]{t}}} \cdot \left(2 \cdot \frac{\cos k}{k \cdot \sin k}\right)\right)\]

25. Applied *-un-lft-identity 7.7

    \[\leadsto \frac{\frac{\frac{1}{\frac{t}{\ell}}}{1}}{k} \cdot \left(\frac{\frac{\frac{\color{blue}{1 \cdot 1}}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{1} \cdot \frac{\sqrt[3]{t}}{\ell}}}{1 \cdot \sin k}}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{1}{\sqrt[3]{t}}} \cdot \left(2 \cdot \frac{\cos k}{k \cdot \sin k}\right)\right)\]

26. Applied times-frac 7.7

    \[\leadsto \frac{\frac{\frac{1}{\frac{t}{\ell}}}{1}}{k} \cdot \left(\frac{\frac{\color{blue}{\frac{1}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{1}} \cdot \frac{1}{\frac{\sqrt[3]{t}}{\ell}}}}{1 \cdot \sin k}}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{1}{\sqrt[3]{t}}} \cdot \left(2 \cdot \frac{\cos k}{k \cdot \sin k}\right)\right)\]

27. Applied times-frac 7.7

    \[\leadsto \frac{\frac{\frac{1}{\frac{t}{\ell}}}{1}}{k} \cdot \left(\frac{\color{blue}{\frac{\frac{1}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{1}}}{1} \cdot \frac{\frac{1}{\frac{\sqrt[3]{t}}{\ell}}}{\sin k}}}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{1}{\sqrt[3]{t}}} \cdot \left(2 \cdot \frac{\cos k}{k \cdot \sin k}\right)\right)\]

28. Applied times-frac 7.7

    \[\leadsto \frac{\frac{\frac{1}{\frac{t}{\ell}}}{1}}{k} \cdot \left(\color{blue}{\left(\frac{\frac{\frac{1}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{1}}}{1}}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \frac{\frac{\frac{1}{\frac{\sqrt[3]{t}}{\ell}}}{\sin k}}{\frac{1}{\sqrt[3]{t}}}\right)} \cdot \left(2 \cdot \frac{\cos k}{k \cdot \sin k}\right)\right)\]

29. Final simplification 7.7

    \[\leadsto \frac{\frac{1}{\frac{t}{\ell}}}{k} \cdot \left(\left(\frac{\frac{\frac{1}{\frac{\sqrt[3]{t}}{\ell}}}{\sin k}}{\frac{1}{\sqrt[3]{t}}} \cdot \frac{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right) \cdot \left(\frac{\cos k}{k \cdot \sin k} \cdot 2\right)\right)\]

Reproduce

herbie shell --seed 2019156 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))))