Toniolo and Linder, Equation (10-)

Average Error: 48.3 → 9.7

Time: 1.8m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le 1.093418808275623190617504932833214890494 \cdot 10^{-47} \lor \neg \left(k \le 4.306881873618754798423481000230201418731 \cdot 10^{117}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\cos k}{\frac{\sin k}{\ell}} \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\sin k}{\ell}}\right) \cdot {\left(\frac{1}{{t}^{1} \cdot {k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\frac{{\left(\sin k\right)}^{2}}{\ell \cdot \ell}} \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right)\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r204508 = 2.0;
        double r204509 = t;
        double r204510 = 3.0;
        double r204511 = pow(r204509, r204510);
        double r204512 = l;
        double r204513 = r204512 * r204512;
        double r204514 = r204511 / r204513;
        double r204515 = k;
        double r204516 = sin(r204515);
        double r204517 = r204514 * r204516;
        double r204518 = tan(r204515);
        double r204519 = r204517 * r204518;
        double r204520 = 1.0;
        double r204521 = r204515 / r204509;
        double r204522 = pow(r204521, r204508);
        double r204523 = r204520 + r204522;
        double r204524 = r204523 - r204520;
        double r204525 = r204519 * r204524;
        double r204526 = r204508 / r204525;
        return r204526;
}

↓

double f(double t, double l, double k) {
        double r204527 = k;
        double r204528 = 1.0934188082756232e-47;
        bool r204529 = r204527 <= r204528;
        double r204530 = 4.306881873618755e+117;
        bool r204531 = r204527 <= r204530;
        double r204532 = !r204531;
        bool r204533 = r204529 || r204532;
        double r204534 = 2.0;
        double r204535 = cos(r204527);
        double r204536 = sin(r204527);
        double r204537 = l;
        double r204538 = r204536 / r204537;
        double r204539 = r204535 / r204538;
        double r204540 = 1.0;
        double r204541 = 2.0;
        double r204542 = r204534 / r204541;
        double r204543 = pow(r204527, r204542);
        double r204544 = r204540 / r204543;
        double r204545 = 1.0;
        double r204546 = pow(r204544, r204545);
        double r204547 = r204546 / r204538;
        double r204548 = r204539 * r204547;
        double r204549 = t;
        double r204550 = pow(r204549, r204545);
        double r204551 = r204550 * r204543;
        double r204552 = r204540 / r204551;
        double r204553 = pow(r204552, r204545);
        double r204554 = r204548 * r204553;
        double r204555 = r204534 * r204554;
        double r204556 = r204540 / r204550;
        double r204557 = pow(r204556, r204545);
        double r204558 = pow(r204536, r204541);
        double r204559 = r204537 * r204537;
        double r204560 = r204558 / r204559;
        double r204561 = r204535 / r204560;
        double r204562 = r204561 * r204546;
        double r204563 = r204557 * r204562;
        double r204564 = r204563 * r204546;
        double r204565 = r204534 * r204564;
        double r204566 = r204533 ? r204555 : r204565;
        return r204566;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if k < 1.0934188082756232e-47 or 4.306881873618755e+117 < k

   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 40.5

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

   3. Taylor expanded around inf 24.0

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

   5. Applied sqr-pow 24.0

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

   6. Applied associate-*r* 21.7

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

   7. Simplified 21.7

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

   9. Applied add-sqr-sqrt 21.7

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

   10. Applied times-frac 21.5

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

   11. Applied unpow-prod-down 21.5

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

   12. Applied associate-*l* 19.9

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

   13. Simplified 19.9

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

   15. Applied sqr-pow 19.9

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

   16. Applied sqr-pow 19.9

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

   17. Applied times-frac 14.9

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

   18. Applied *-un-lft-identity 14.9

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

   19. Applied times-frac 14.8

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

   20. Applied associate-*r* 8.9

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

   21. Simplified 8.8

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

   if 1.0934188082756232e-47 < k < 4.306881873618755e+117

   1. Initial program 52.3

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

   2. Simplified 41.9

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

   3. Taylor expanded around inf 13.7

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

   5. Applied sqr-pow 13.7

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

   6. Applied associate-*r* 13.7

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

   7. Simplified 13.7

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

   9. Applied add-sqr-sqrt 13.7

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

   10. Applied times-frac 13.6

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

   11. Applied unpow-prod-down 13.6

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

   12. Applied associate-*l* 13.5

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

   13. Simplified 13.6

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

   15. Applied add-cube-cbrt 13.6

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

   16. Applied sqrt-prod 13.6

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

   17. Applied times-frac 13.5

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

   18. Applied unpow-prod-down 13.5

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

   19. Applied associate-*l* 14.0

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

   20. Simplified 14.0

       \[\leadsto 2 \cdot \left({\left(\frac{\sqrt{\sqrt[3]{1} \cdot \sqrt[3]{1}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \color{blue}{\left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{\cos k}{\frac{{\left(\sin k\right)}^{2}}{\ell \cdot \ell}}\right) \cdot {\left(\frac{1}{{t}^{1}}\right)}^{1}\right)}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 1.093418808275623190617504932833214890494 \cdot 10^{-47} \lor \neg \left(k \le 4.306881873618754798423481000230201418731 \cdot 10^{117}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\cos k}{\frac{\sin k}{\ell}} \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\sin k}{\ell}}\right) \cdot {\left(\frac{1}{{t}^{1} \cdot {k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\frac{{\left(\sin k\right)}^{2}}{\ell \cdot \ell}} \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right)\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))