Toniolo and Linder, Equation (10-)

Average Error: 47.1 → 11.5

Time: 5.2m

Precision: 64

Math

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

↓

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

C

double f(double t, double l, double k) {
        double r10671387 = 2.0;
        double r10671388 = t;
        double r10671389 = 3.0;
        double r10671390 = pow(r10671388, r10671389);
        double r10671391 = l;
        double r10671392 = r10671391 * r10671391;
        double r10671393 = r10671390 / r10671392;
        double r10671394 = k;
        double r10671395 = sin(r10671394);
        double r10671396 = r10671393 * r10671395;
        double r10671397 = tan(r10671394);
        double r10671398 = r10671396 * r10671397;
        double r10671399 = 1.0;
        double r10671400 = r10671394 / r10671388;
        double r10671401 = pow(r10671400, r10671387);
        double r10671402 = r10671399 + r10671401;
        double r10671403 = r10671402 - r10671399;
        double r10671404 = r10671398 * r10671403;
        double r10671405 = r10671387 / r10671404;
        return r10671405;
}

↓

double f(double t, double l, double k) {
        double r10671406 = 2.0;
        double r10671407 = cbrt(r10671406);
        double r10671408 = sqrt(r10671407);
        double r10671409 = 1.0;
        double r10671410 = l;
        double r10671411 = t;
        double r10671412 = r10671410 / r10671411;
        double r10671413 = cbrt(r10671412);
        double r10671414 = r10671413 * r10671413;
        double r10671415 = r10671409 / r10671414;
        double r10671416 = r10671408 / r10671415;
        double r10671417 = k;
        double r10671418 = r10671417 / r10671411;
        double r10671419 = cbrt(r10671418);
        double r10671420 = r10671419 * r10671419;
        double r10671421 = r10671416 / r10671420;
        double r10671422 = cbrt(r10671411);
        double r10671423 = r10671422 / r10671413;
        double r10671424 = r10671408 / r10671423;
        double r10671425 = sin(r10671417);
        double r10671426 = r10671424 / r10671425;
        double r10671427 = r10671426 / r10671419;
        double r10671428 = r10671421 * r10671427;
        double r10671429 = tan(r10671417);
        double r10671430 = r10671412 / r10671429;
        double r10671431 = r10671407 / r10671422;
        double r10671432 = r10671431 * r10671431;
        double r10671433 = r10671432 / r10671418;
        double r10671434 = r10671430 * r10671433;
        double r10671435 = r10671428 * r10671434;
        return r10671435;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Initial program 47.1

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

2. Simplified 30.6

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

4. Applied add-cube-cbrt 30.7

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

5. Applied times-frac 30.2

   \[\leadsto \frac{\frac{\frac{2}{\color{blue}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\frac{\ell}{t}} \cdot \frac{\sqrt[3]{t}}{\frac{\ell}{t}}}}}{\tan k \cdot \sin k}}{\frac{k}{t} \cdot \frac{k}{t}}\]

6. Applied add-cube-cbrt 30.3

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

7. Applied times-frac 30.2

   \[\leadsto \frac{\frac{\color{blue}{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\frac{\ell}{t}}} \cdot \frac{\sqrt[3]{2}}{\frac{\sqrt[3]{t}}{\frac{\ell}{t}}}}}{\tan k \cdot \sin k}}{\frac{k}{t} \cdot \frac{k}{t}}\]

8. Applied times-frac 29.5

   \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\frac{\ell}{t}}}}{\tan k} \cdot \frac{\frac{\sqrt[3]{2}}{\frac{\sqrt[3]{t}}{\frac{\ell}{t}}}}{\sin k}}}{\frac{k}{t} \cdot \frac{k}{t}}\]

9. Applied times-frac 15.2

   \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\frac{\ell}{t}}}}{\tan k}}{\frac{k}{t}} \cdot \frac{\frac{\frac{\sqrt[3]{2}}{\frac{\sqrt[3]{t}}{\frac{\ell}{t}}}}{\sin k}}{\frac{k}{t}}}\]

10. Simplified 12.8

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

12. Applied add-cube-cbrt 12.9

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

13. Applied *-un-lft-identity 12.9

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

14. Applied add-cube-cbrt 12.9

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

15. Applied *-un-lft-identity 12.9

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

16. Applied times-frac 12.9

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

17. Applied add-sqr-sqrt 12.8

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

18. Applied times-frac 12.7

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

19. Applied times-frac 12.3

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

20. Applied times-frac 11.5

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

21. Final simplification 11.5

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

Reproduce

herbie shell --seed 2019168 
(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))))