Toniolo and Linder, Equation (10-)

Average Error: 48.1 → 14.0

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 1.084960617494571136532106729940016148301 \cdot 10^{-276}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \mathsf{fma}\left(\frac{\ell}{k}, \frac{\ell}{k}, {\ell}^{2} \cdot \frac{-1}{6}\right)\right)\\ \mathbf{elif}\;\ell \cdot \ell \le 1.064294734521519693928254286704808478679 \cdot 10^{294}:\\ \;\;\;\;\left(\left(\frac{{\ell}^{2}}{\sin k} \cdot \left(\frac{\cos k}{\sin k} \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1}\right)\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\frac{{\left(\sqrt{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt{t}\right)}^{3}}{\ell}\right) \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r101539 = 2.0;
        double r101540 = t;
        double r101541 = 3.0;
        double r101542 = pow(r101540, r101541);
        double r101543 = l;
        double r101544 = r101543 * r101543;
        double r101545 = r101542 / r101544;
        double r101546 = k;
        double r101547 = sin(r101546);
        double r101548 = r101545 * r101547;
        double r101549 = tan(r101546);
        double r101550 = r101548 * r101549;
        double r101551 = 1.0;
        double r101552 = r101546 / r101540;
        double r101553 = pow(r101552, r101539);
        double r101554 = r101551 + r101553;
        double r101555 = r101554 - r101551;
        double r101556 = r101550 * r101555;
        double r101557 = r101539 / r101556;
        return r101557;
}

↓

double f(double t, double l, double k) {
        double r101558 = l;
        double r101559 = r101558 * r101558;
        double r101560 = 1.0849606174945711e-276;
        bool r101561 = r101559 <= r101560;
        double r101562 = 2.0;
        double r101563 = 1.0;
        double r101564 = k;
        double r101565 = 2.0;
        double r101566 = r101562 / r101565;
        double r101567 = pow(r101564, r101566);
        double r101568 = t;
        double r101569 = 1.0;
        double r101570 = pow(r101568, r101569);
        double r101571 = r101567 * r101570;
        double r101572 = r101567 * r101571;
        double r101573 = r101563 / r101572;
        double r101574 = pow(r101573, r101569);
        double r101575 = r101558 / r101564;
        double r101576 = pow(r101558, r101565);
        double r101577 = -0.16666666666666666;
        double r101578 = r101576 * r101577;
        double r101579 = fma(r101575, r101575, r101578);
        double r101580 = r101574 * r101579;
        double r101581 = r101562 * r101580;
        double r101582 = 1.0642947345215197e+294;
        bool r101583 = r101559 <= r101582;
        double r101584 = sin(r101564);
        double r101585 = r101576 / r101584;
        double r101586 = cos(r101564);
        double r101587 = r101586 / r101584;
        double r101588 = r101563 / r101571;
        double r101589 = pow(r101588, r101569);
        double r101590 = r101587 * r101589;
        double r101591 = r101585 * r101590;
        double r101592 = r101563 / r101567;
        double r101593 = pow(r101592, r101569);
        double r101594 = r101591 * r101593;
        double r101595 = r101594 * r101562;
        double r101596 = sqrt(r101568);
        double r101597 = 3.0;
        double r101598 = pow(r101596, r101597);
        double r101599 = r101598 / r101558;
        double r101600 = r101599 * r101599;
        double r101601 = r101600 * r101584;
        double r101602 = tan(r101564);
        double r101603 = r101601 * r101602;
        double r101604 = r101562 / r101603;
        double r101605 = r101564 / r101568;
        double r101606 = pow(r101605, r101562);
        double r101607 = r101604 / r101606;
        double r101608 = r101583 ? r101595 : r101607;
        double r101609 = r101561 ? r101581 : r101608;
        return r101609;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if (* l l) < 1.0849606174945711e-276

   1. Initial program 45.5

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

   2. Simplified 36.7

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

   3. Taylor expanded around inf 18.4

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

   5. Applied sqr-pow 18.4

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

   6. Applied associate-*l* 18.4

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

   7. Taylor expanded around 0 19.0

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} - \frac{1}{6} \cdot {\ell}^{2}\right)}\right)\]

   8. Simplified 10.1

      \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \color{blue}{\mathsf{fma}\left(\frac{\ell}{k}, \frac{\ell}{k}, {\ell}^{2} \cdot \frac{-1}{6}\right)}\right)\]

   if 1.0849606174945711e-276 < (* l l) < 1.0642947345215197e+294

   1. Initial program 44.7

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

   2. Simplified 35.6

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

   3. Taylor expanded around inf 10.9

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

   5. Applied sqr-pow 10.9

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

   6. Applied associate-*l* 6.7

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

   8. Applied *-un-lft-identity 6.7

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

   9. Applied times-frac 6.3

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

   10. Applied unpow-prod-down 6.3

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

   11. Applied associate-*l* 3.5

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

   13. Applied sqr-pow 3.5

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

   14. Applied times-frac 3.1

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

   15. Applied associate-*r* 3.0

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

   16. Simplified 3.0

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

   if 1.0642947345215197e+294 < (* l l)

   1. Initial program 63.3

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

   2. Simplified 63.1

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

   4. Applied add-sqr-sqrt 63.6

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

   5. Applied unpow-prod-down 63.6

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

   6. Applied times-frac 53.5

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

4. Final simplification 14.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 1.084960617494571136532106729940016148301 \cdot 10^{-276}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \mathsf{fma}\left(\frac{\ell}{k}, \frac{\ell}{k}, {\ell}^{2} \cdot \frac{-1}{6}\right)\right)\\ \mathbf{elif}\;\ell \cdot \ell \le 1.064294734521519693928254286704808478679 \cdot 10^{294}:\\ \;\;\;\;\left(\left(\frac{{\ell}^{2}}{\sin k} \cdot \left(\frac{\cos k}{\sin k} \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1}\right)\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\frac{{\left(\sqrt{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt{t}\right)}^{3}}{\ell}\right) \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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))))