Toniolo and Linder, Equation (10+)

Average Error: 32.4 → 25.3

Time: 16.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -8.19471004965374614 \cdot 10^{-96}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{elif}\;t \le 4.42712960801617489 \cdot 10^{-84}:\\ \;\;\;\;\mathsf{fma}\left(0.23333333333333334, \frac{t \cdot {\ell}^{2}}{{k}^{2}}, -\mathsf{fma}\left(0.11666666666666667, {\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot {\ell}^{2}, 0.333333333333333315 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot {\left(\frac{1}{{t}^{1}}\right)}^{1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}} \cdot \left(\frac{\frac{2 \cdot \ell}{{t}^{3} \cdot \sin k}}{\sqrt[3]{\tan k}} \cdot \left(\sqrt[3]{\frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}} \cdot \sqrt[3]{\frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\right)\right)\right) \cdot \sqrt[3]{\frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r114490 = 2.0;
        double r114491 = t;
        double r114492 = 3.0;
        double r114493 = pow(r114491, r114492);
        double r114494 = l;
        double r114495 = r114494 * r114494;
        double r114496 = r114493 / r114495;
        double r114497 = k;
        double r114498 = sin(r114497);
        double r114499 = r114496 * r114498;
        double r114500 = tan(r114497);
        double r114501 = r114499 * r114500;
        double r114502 = 1.0;
        double r114503 = r114497 / r114491;
        double r114504 = pow(r114503, r114490);
        double r114505 = r114502 + r114504;
        double r114506 = r114505 + r114502;
        double r114507 = r114501 * r114506;
        double r114508 = r114490 / r114507;
        return r114508;
}

↓

double f(double t, double l, double k) {
        double r114509 = t;
        double r114510 = -8.194710049653746e-96;
        bool r114511 = r114509 <= r114510;
        double r114512 = 2.0;
        double r114513 = l;
        double r114514 = r114512 * r114513;
        double r114515 = 3.0;
        double r114516 = pow(r114509, r114515);
        double r114517 = k;
        double r114518 = sin(r114517);
        double r114519 = r114516 * r114518;
        double r114520 = r114514 / r114519;
        double r114521 = tan(r114517);
        double r114522 = r114520 / r114521;
        double r114523 = 2.0;
        double r114524 = 1.0;
        double r114525 = r114517 / r114509;
        double r114526 = pow(r114525, r114512);
        double r114527 = fma(r114523, r114524, r114526);
        double r114528 = r114513 / r114527;
        double r114529 = r114522 * r114528;
        double r114530 = 4.427129608016175e-84;
        bool r114531 = r114509 <= r114530;
        double r114532 = 0.23333333333333334;
        double r114533 = pow(r114513, r114523);
        double r114534 = r114509 * r114533;
        double r114535 = pow(r114517, r114523);
        double r114536 = r114534 / r114535;
        double r114537 = 0.11666666666666667;
        double r114538 = 1.0;
        double r114539 = pow(r114509, r114524);
        double r114540 = r114538 / r114539;
        double r114541 = pow(r114540, r114524);
        double r114542 = r114541 * r114533;
        double r114543 = 0.3333333333333333;
        double r114544 = r114533 / r114535;
        double r114545 = r114544 * r114541;
        double r114546 = r114543 * r114545;
        double r114547 = fma(r114537, r114542, r114546);
        double r114548 = -r114547;
        double r114549 = fma(r114532, r114536, r114548);
        double r114550 = cbrt(r114521);
        double r114551 = r114550 * r114550;
        double r114552 = r114538 / r114551;
        double r114553 = r114520 / r114550;
        double r114554 = cbrt(r114528);
        double r114555 = r114554 * r114554;
        double r114556 = r114553 * r114555;
        double r114557 = r114552 * r114556;
        double r114558 = r114557 * r114554;
        double r114559 = r114531 ? r114549 : r114558;
        double r114560 = r114511 ? r114529 : r114559;
        return r114560;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if t < -8.194710049653746e-96

   1. Initial program 22.7

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

   2. Simplified 22.3

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

   4. Applied *-un-lft-identity 22.3

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

   5. Applied times-frac 21.1

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

   6. Applied associate-*r* 18.3

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

   7. Simplified 16.8

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

   9. Applied associate-*l/ 16.8

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

   if -8.194710049653746e-96 < t < 4.427129608016175e-84

   1. Initial program 60.5

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

   2. Simplified 61.4

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

   4. Applied *-un-lft-identity 61.4

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

   5. Applied times-frac 61.4

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

   6. Applied associate-*r* 61.7

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

   7. Simplified 61.5

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

   9. Applied associate-*l/ 60.9

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

   10. Taylor expanded around 0 50.2

       \[\leadsto \color{blue}{0.23333333333333334 \cdot \frac{t \cdot {\ell}^{2}}{{k}^{2}} - \left(0.11666666666666667 \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot {\ell}^{2}\right) + 0.333333333333333315 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot {\left(\frac{1}{{t}^{1}}\right)}^{1}\right)\right)}\]

   11. Simplified 50.2

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.23333333333333334, \frac{t \cdot {\ell}^{2}}{{k}^{2}}, -\mathsf{fma}\left(0.11666666666666667, {\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot {\ell}^{2}, 0.333333333333333315 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot {\left(\frac{1}{{t}^{1}}\right)}^{1}\right)\right)\right)}\]

   if 4.427129608016175e-84 < t

   1. Initial program 23.8

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

   2. Simplified 23.8

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

   4. Applied *-un-lft-identity 23.8

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

   5. Applied times-frac 22.7

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

   6. Applied associate-*r* 19.4

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

   7. Simplified 17.7

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

   9. Applied associate-*l/ 17.7

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

   11. Applied add-cube-cbrt 17.8

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

   12. Applied associate-*r* 17.8

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

   14. Applied add-cube-cbrt 17.9

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

   15. Applied *-un-lft-identity 17.9

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

   16. Applied times-frac 17.9

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

   17. Applied associate-*l* 17.5

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

4. Final simplification 25.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -8.19471004965374614 \cdot 10^{-96}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{elif}\;t \le 4.42712960801617489 \cdot 10^{-84}:\\ \;\;\;\;\mathsf{fma}\left(0.23333333333333334, \frac{t \cdot {\ell}^{2}}{{k}^{2}}, -\mathsf{fma}\left(0.11666666666666667, {\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot {\ell}^{2}, 0.333333333333333315 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot {\left(\frac{1}{{t}^{1}}\right)}^{1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}} \cdot \left(\frac{\frac{2 \cdot \ell}{{t}^{3} \cdot \sin k}}{\sqrt[3]{\tan k}} \cdot \left(\sqrt[3]{\frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}} \cdot \sqrt[3]{\frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\right)\right)\right) \cdot \sqrt[3]{\frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +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))))