Average Error: 32.3 → 11.4
Time: 45.3s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\frac{1}{\left(\sin k \cdot t\right) \cdot \frac{t \cdot \sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}}}{\ell}} \cdot \frac{\frac{2}{\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}}}}{\tan k \cdot \frac{t}{\ell}}\]
\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{1}{\left(\sin k \cdot t\right) \cdot \frac{t \cdot \sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}}}{\ell}} \cdot \frac{\frac{2}{\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}}}}{\tan k \cdot \frac{t}{\ell}}
double f(double t, double l, double k) {
        double r2888534 = 2.0;
        double r2888535 = t;
        double r2888536 = 3.0;
        double r2888537 = pow(r2888535, r2888536);
        double r2888538 = l;
        double r2888539 = r2888538 * r2888538;
        double r2888540 = r2888537 / r2888539;
        double r2888541 = k;
        double r2888542 = sin(r2888541);
        double r2888543 = r2888540 * r2888542;
        double r2888544 = tan(r2888541);
        double r2888545 = r2888543 * r2888544;
        double r2888546 = 1.0;
        double r2888547 = r2888541 / r2888535;
        double r2888548 = pow(r2888547, r2888534);
        double r2888549 = r2888546 + r2888548;
        double r2888550 = r2888549 + r2888546;
        double r2888551 = r2888545 * r2888550;
        double r2888552 = r2888534 / r2888551;
        return r2888552;
}
↓
double f(double t, double l, double k) {
        double r2888553 = 1.0;
        double r2888554 = k;
        double r2888555 = sin(r2888554);
        double r2888556 = t;
        double r2888557 = r2888555 * r2888556;
        double r2888558 = 2.0;
        double r2888559 = r2888554 / r2888556;
        double r2888560 = r2888559 * r2888559;
        double r2888561 = r2888558 + r2888560;
        double r2888562 = sqrt(r2888561);
        double r2888563 = r2888556 * r2888562;
        double r2888564 = l;
        double r2888565 = r2888563 / r2888564;
        double r2888566 = r2888557 * r2888565;
        double r2888567 = r2888553 / r2888566;
        double r2888568 = r2888558 / r2888562;
        double r2888569 = tan(r2888554);
        double r2888570 = r2888556 / r2888564;
        double r2888571 = r2888569 * r2888570;
        double r2888572 = r2888568 / r2888571;
        double r2888573 = r2888567 * r2888572;
        return r2888573;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.3

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

  2. Simplified17.0

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

  4. Applied add-sqr-sqrt17.1

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

  5. Applied *-un-lft-identity17.1

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

  6. Applied times-frac17.1

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

  7. Applied times-frac15.0

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

  9. Applied *-un-lft-identity15.0

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

  10. Applied associate-/l*15.0

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

  11. Simplified12.6

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

  13. Applied add-sqr-sqrt12.5

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

  15. Applied associate-*r*11.4

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

  16. Simplified11.4

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

  17. Final simplification11.4

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

Reproduce

herbie shell --seed 2019135 
(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))))