Average Error: 48.4 → 6.5
Time: 27.1s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 1.3923082163925979 \cdot 10^{-149} \lor \neg \left(\ell \cdot \ell \leq 3.6894671702728764 \cdot 10^{+128}\right):\\ \;\;\;\;2 \cdot \frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \cos k}{\sin k \cdot \left(t \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\left|\frac{\ell}{k}\right| \cdot \left|\ell\right|\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \left|k\right|}\\ \end{array}\]
\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}\;\ell \cdot \ell \leq 1.3923082163925979 \cdot 10^{-149} \lor \neg \left(\ell \cdot \ell \leq 3.6894671702728764 \cdot 10^{+128}\right):\\
\;\;\;\;2 \cdot \frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \cos k}{\sin k \cdot \left(t \cdot \sin k\right)}\\

\mathbf{else}:\\
\;\;\;\;2 \cdot \frac{\cos k \cdot \left(\left|\frac{\ell}{k}\right| \cdot \left|\ell\right|\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \left|k\right|}\\

\end{array}
(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
 (if (or (<= (* l l) 1.3923082163925979e-149)
         (not (<= (* l l) 3.6894671702728764e+128)))
   (* 2.0 (/ (* (* (/ l k) (/ l k)) (cos k)) (* (sin k) (* t (sin k)))))
   (*
    2.0
    (/
     (* (cos k) (* (fabs (/ l k)) (fabs l)))
     (* (* t (pow (sin k) 2.0)) (fabs k))))))
double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0));
}
↓
double code(double t, double l, double k) {
	double tmp;
	if (((l * l) <= 1.3923082163925979e-149) || !((l * l) <= 3.6894671702728764e+128)) {
		tmp = 2.0 * ((((l / k) * (l / k)) * cos(k)) / (sin(k) * (t * sin(k))));
	} else {
		tmp = 2.0 * ((cos(k) * (fabs(l / k) * fabs(l))) / ((t * pow(sin(k), 2.0)) * fabs(k)));
	}
	return tmp;
}

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. Split input into 2 regimes

  2. if (*.f64 l l) < 1.3923082163925979e-149 or 3.6894671702728764e128 < (*.f64 l l)

    1. Initial program 49.8

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

    2. Simplified42.6

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

    3. Taylor expanded around 0 26.6

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

    4. Simplified26.6

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

    6. Applied associate-/r*_binary6425.7

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

    7. Simplified25.7

      \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell \cdot \ell}{k \cdot k} \cdot \cos k}}{t \cdot {\sin k}^{2}}\]
    8. Using strategy rm

    9. Applied add-sqr-sqrt_binary6425.8

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

    10. Simplified25.7

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

    11. Simplified10.1

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

    13. Applied unpow2_binary6410.1

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

    14. Applied associate-*r*_binary648.2

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

    if 1.3923082163925979e-149 < (*.f64 l l) < 3.6894671702728764e128

    1. Initial program 43.7

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

    2. Simplified33.8

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

    3. Taylor expanded around 0 8.8

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

    4. Simplified8.8

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

    6. Applied associate-/r*_binary647.6

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

    7. Simplified7.6

      \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell \cdot \ell}{k \cdot k} \cdot \cos k}}{t \cdot {\sin k}^{2}}\]
    8. Using strategy rm

    9. Applied add-sqr-sqrt_binary647.7

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

    10. Simplified7.6

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

    11. Simplified6.3

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

    13. Applied fabs-div_binary646.3

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

    14. Applied associate-*l/_binary646.2

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

    15. Applied associate-*l/_binary646.2

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

    16. Applied associate-/l/_binary641.1

      \[\leadsto 2 \cdot \color{blue}{\frac{\left(\left|\ell\right| \cdot \left|\frac{\ell}{k}\right|\right) \cdot \cos k}{\left(t \cdot {\sin k}^{2}\right) \cdot \left|k\right|}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 1.3923082163925979 \cdot 10^{-149} \lor \neg \left(\ell \cdot \ell \leq 3.6894671702728764 \cdot 10^{+128}\right):\\ \;\;\;\;2 \cdot \frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \cos k}{\sin k \cdot \left(t \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\left|\frac{\ell}{k}\right| \cdot \left|\ell\right|\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \left|k\right|}\\ \end{array}\]

Reproduce

herbie shell --seed 2021174 
(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))))