Average Error: 47.9 → 10.7
Time: 1.3m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\cos k}{\frac{\sin k}{\ell} \cdot \sin k} \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}} \cdot \frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot 2\right) \cdot \ell\right)\]
\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{\cos k}{\frac{\sin k}{\ell} \cdot \sin k} \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}} \cdot \frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot 2\right) \cdot \ell\right)
double f(double t, double l, double k) {
        double r9889530 = 2.0;
        double r9889531 = t;
        double r9889532 = 3.0;
        double r9889533 = pow(r9889531, r9889532);
        double r9889534 = l;
        double r9889535 = r9889534 * r9889534;
        double r9889536 = r9889533 / r9889535;
        double r9889537 = k;
        double r9889538 = sin(r9889537);
        double r9889539 = r9889536 * r9889538;
        double r9889540 = tan(r9889537);
        double r9889541 = r9889539 * r9889540;
        double r9889542 = 1.0;
        double r9889543 = r9889537 / r9889531;
        double r9889544 = pow(r9889543, r9889530);
        double r9889545 = r9889542 + r9889544;
        double r9889546 = r9889545 - r9889542;
        double r9889547 = r9889541 * r9889546;
        double r9889548 = r9889530 / r9889547;
        return r9889548;
}
↓
double f(double t, double l, double k) {
        double r9889549 = k;
        double r9889550 = cos(r9889549);
        double r9889551 = sin(r9889549);
        double r9889552 = l;
        double r9889553 = r9889551 / r9889552;
        double r9889554 = r9889553 * r9889551;
        double r9889555 = r9889550 / r9889554;
        double r9889556 = 1.0;
        double r9889557 = 2.0;
        double r9889558 = 2.0;
        double r9889559 = r9889557 / r9889558;
        double r9889560 = pow(r9889549, r9889559);
        double r9889561 = t;
        double r9889562 = 1.0;
        double r9889563 = pow(r9889561, r9889562);
        double r9889564 = r9889560 * r9889563;
        double r9889565 = r9889556 / r9889564;
        double r9889566 = r9889556 / r9889560;
        double r9889567 = r9889565 * r9889566;
        double r9889568 = pow(r9889567, r9889562);
        double r9889569 = r9889568 * r9889557;
        double r9889570 = r9889569 * r9889552;
        double r9889571 = r9889555 * r9889570;
        return r9889571;
}

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 47.9

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

  2. Simplified40.8

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

  3. Taylor expanded around inf 22.4

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

  4. Simplified20.5

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

  6. Applied associate-*r/20.4

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

  7. Applied associate-/r/20.3

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

  8. Applied associate-*l*15.7

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

  10. Applied sqr-pow15.7

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

  11. Applied associate-*r*11.0

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

  13. Applied *-un-lft-identity11.0

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

  14. Applied times-frac10.7

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

  15. Final simplification10.7

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

Reproduce

herbie shell --seed 2019172 
(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))))