Average Error: 31.4 → 7.0
Time: 44.7s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\frac{2}{\frac{\mathsf{fma}\left(2, \frac{\frac{\sin k \cdot t}{\frac{\ell}{\sin k \cdot t}}}{\cos k}, \frac{\sin k \cdot k}{\ell} \cdot \frac{\sin k \cdot k}{\cos k}\right)}{\frac{\ell}{t}}}\]
\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{2}{\frac{\mathsf{fma}\left(2, \frac{\frac{\sin k \cdot t}{\frac{\ell}{\sin k \cdot t}}}{\cos k}, \frac{\sin k \cdot k}{\ell} \cdot \frac{\sin k \cdot k}{\cos k}\right)}{\frac{\ell}{t}}}
double f(double t, double l, double k) {
        double r2841506 = 2.0;
        double r2841507 = t;
        double r2841508 = 3.0;
        double r2841509 = pow(r2841507, r2841508);
        double r2841510 = l;
        double r2841511 = r2841510 * r2841510;
        double r2841512 = r2841509 / r2841511;
        double r2841513 = k;
        double r2841514 = sin(r2841513);
        double r2841515 = r2841512 * r2841514;
        double r2841516 = tan(r2841513);
        double r2841517 = r2841515 * r2841516;
        double r2841518 = 1.0;
        double r2841519 = r2841513 / r2841507;
        double r2841520 = pow(r2841519, r2841506);
        double r2841521 = r2841518 + r2841520;
        double r2841522 = r2841521 + r2841518;
        double r2841523 = r2841517 * r2841522;
        double r2841524 = r2841506 / r2841523;
        return r2841524;
}
↓
double f(double t, double l, double k) {
        double r2841525 = 2.0;
        double r2841526 = k;
        double r2841527 = sin(r2841526);
        double r2841528 = t;
        double r2841529 = r2841527 * r2841528;
        double r2841530 = l;
        double r2841531 = r2841530 / r2841529;
        double r2841532 = r2841529 / r2841531;
        double r2841533 = cos(r2841526);
        double r2841534 = r2841532 / r2841533;
        double r2841535 = r2841527 * r2841526;
        double r2841536 = r2841535 / r2841530;
        double r2841537 = r2841535 / r2841533;
        double r2841538 = r2841536 * r2841537;
        double r2841539 = fma(r2841525, r2841534, r2841538);
        double r2841540 = r2841530 / r2841528;
        double r2841541 = r2841539 / r2841540;
        double r2841542 = r2841525 / r2841541;
        return r2841542;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Initial program 31.4

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

  2. Simplified17.4

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

  4. Applied associate-*l/16.5

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

  5. Applied associate-*l/14.9

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

  6. Applied associate-*l/13.6

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

  7. Taylor expanded around inf 20.7

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

  8. Simplified13.1

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

  10. Applied associate-/l*10.1

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

  12. Applied times-frac7.0

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

  13. Final simplification7.0

    \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(2, \frac{\frac{\sin k \cdot t}{\frac{\ell}{\sin k \cdot t}}}{\cos k}, \frac{\sin k \cdot k}{\ell} \cdot \frac{\sin k \cdot k}{\cos k}\right)}{\frac{\ell}{t}}}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(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))))