Average Error: 48.0 → 8.4
Time: 29.1s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\ell \cdot \left(2 \cdot \left({\left({\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1}\right)}^{1} \cdot \left(\left(\ell \cdot \frac{\cos k}{{\left(\sin k\right)}^{2}}\right) \cdot {\left({\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1} \cdot {\left({t}^{\left(-1\right)}\right)}^{1}\right)}^{1}\right)\right)\right)\]
\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)}
↓
\ell \cdot \left(2 \cdot \left({\left({\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1}\right)}^{1} \cdot \left(\left(\ell \cdot \frac{\cos k}{{\left(\sin k\right)}^{2}}\right) \cdot {\left({\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1} \cdot {\left({t}^{\left(-1\right)}\right)}^{1}\right)}^{1}\right)\right)\right)
(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))
↓
(FPCore (t l k)
 :precision binary64
 (*
  l
  (*
   2.0
   (*
    (pow (pow (pow k (/ (- 2.0) 2.0)) 1.0) 1.0)
    (*
     (* l (/ (cos k) (pow (sin k) 2.0)))
     (pow
      (* (pow (pow k (/ (- 2.0) 2.0)) 1.0) (pow (pow t (- 1.0)) 1.0))
      1.0))))))
double code(double t, double l, double k) {
	return (2.0 / ((double) (((double) (((double) ((((double) pow(t, 3.0)) / ((double) (l * l))) * ((double) sin(k)))) * ((double) tan(k)))) * ((double) (((double) (1.0 + ((double) pow((k / t), 2.0)))) - 1.0)))));
}
↓
double code(double t, double l, double k) {
	return ((double) (l * ((double) (2.0 * ((double) (((double) pow(((double) pow(((double) pow(k, (((double) -(2.0)) / 2.0))), 1.0)), 1.0)) * ((double) (((double) (l * (((double) cos(k)) / ((double) pow(((double) sin(k)), 2.0))))) * ((double) pow(((double) (((double) pow(((double) pow(k, (((double) -(2.0)) / 2.0))), 1.0)) * ((double) pow(((double) pow(t, ((double) -(1.0)))), 1.0)))), 1.0))))))))));
}

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 Error: 48.0 bits

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

  2. SimplifiedError: 38.7 bits

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

  3. Taylor expanded around inf Error: 52.5 bits

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

  4. SimplifiedError: 16.0 bits

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

  6. Applied sqr-powError: 16.1 bits

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

  7. Applied unpow-prod-downError: 16.1 bits

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

  8. Applied associate-*l*Error: 12.9 bits

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

  9. SimplifiedError: 12.9 bits

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

  11. Applied unpow-prod-downError: 12.9 bits

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

  12. Applied associate-*l*Error: 8.4 bits

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

  13. SimplifiedError: 9.4 bits

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

  15. Applied associate-*r*Error: 8.4 bits

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

  16. Final simplificationError: 8.4 bits

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

Reproduce

herbie shell --seed 2020203 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))