Average Error: 48.6 → 6.9
Time: 30.4s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4.908952939768733 \cdot 10^{-257} \lor \neg \left(\ell \cdot \ell \leq 6.352659668616961 \cdot 10^{+286}\right):\\ \;\;\;\;2 \cdot \frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \cos k}{\sin k \cdot \left(t \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \frac{\ell}{k}\right)}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array}\]
\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 4.908952939768733 \cdot 10^{-257} \lor \neg \left(\ell \cdot \ell \leq 6.352659668616961 \cdot 10^{+286}\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(\ell \cdot \frac{\ell}{k}\right)}{k \cdot \left(t \cdot {\sin k}^{2}\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) 4.908952939768733e-257)
         (not (<= (* l l) 6.352659668616961e+286)))
   (* 2.0 (/ (* (* (/ l k) (/ l k)) (cos k)) (* (sin k) (* t (sin k)))))
   (* 2.0 (/ (* (cos k) (* l (/ l k))) (* k (* t (pow (sin k) 2.0)))))))
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) <= 4.908952939768733e-257) || !((l * l) <= 6.352659668616961e+286)) {
		tmp = 2.0 * ((((l / k) * (l / k)) * cos(k)) / (sin(k) * (t * sin(k))));
	} else {
		tmp = 2.0 * ((cos(k) * (l * (l / k))) / (k * (t * pow(sin(k), 2.0))));
	}
	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) < 4.908952939768733e-257 or 6.3526596686169608e286 < (*.f64 l l)

    1. Initial program 51.6

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

    2. Simplified44.8

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

    3. Taylor expanded around 0 32.1

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

    4. Simplified32.1

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

    6. Applied associate-/r*_binary64_36331.4

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

    7. Simplified31.4

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

    9. Applied times-frac_binary64_42512.6

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

    11. Applied unpow2_binary64_48412.6

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

    12. Applied associate-*r*_binary64_3599.8

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

    if 4.908952939768733e-257 < (*.f64 l l) < 6.3526596686169608e286

    1. Initial program 45.0

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

    2. Simplified36.1

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

    3. Taylor expanded around 0 12.9

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

    4. Simplified12.9

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

    6. Applied associate-/r*_binary64_36311.2

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

    7. Simplified11.2

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

    9. Applied times-frac_binary64_4256.8

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

    11. Applied associate-*l/_binary64_3626.7

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

    12. Applied associate-*l/_binary64_3626.7

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

    13. Applied associate-/l/_binary64_3663.2

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

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4.908952939768733 \cdot 10^{-257} \lor \neg \left(\ell \cdot \ell \leq 6.352659668616961 \cdot 10^{+286}\right):\\ \;\;\;\;2 \cdot \frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \cos k}{\sin k \cdot \left(t \cdot \sin k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \frac{\ell}{k}\right)}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021019 
(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))))