Average Error: 32.7 → 13.3
Time: 10.2s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -2.882279254698786 \cdot 10^{-232} \lor \neg \left(t \leq 1.4895900323393487 \cdot 10^{-174}\right):\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \ell\right) \cdot \cos k} + 2 \cdot \frac{{\sin k}^{2} \cdot {t}^{3}}{\left(\ell \cdot \ell\right) \cdot \cos k}}\\ \end{array}\]
\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}\;t \leq -2.882279254698786 \cdot 10^{-232} \lor \neg \left(t \leq 1.4895900323393487 \cdot 10^{-174}\right):\\
\;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \ell\right) \cdot \cos k} + 2 \cdot \frac{{\sin k}^{2} \cdot {t}^{3}}{\left(\ell \cdot \ell\right) \cdot \cos k}}\\

\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 (<= t -2.882279254698786e-232) (not (<= t 1.4895900323393487e-174)))
   (/
    2.0
    (*
     (/ t l)
     (* (* (* (/ t l) (* t (sin k))) (tan k)) (+ 2.0 (pow (/ k t) 2.0)))))
   (/
    2.0
    (+
     (/ (* (* k k) (* t (pow (sin k) 2.0))) (* (* l l) (cos k)))
     (* 2.0 (/ (* (pow (sin k) 2.0) (pow t 3.0)) (* (* l l) (cos 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 ((t <= -2.882279254698786e-232) || !(t <= 1.4895900323393487e-174)) {
		tmp = 2.0 / ((t / l) * ((((t / l) * (t * sin(k))) * tan(k)) * (2.0 + pow((k / t), 2.0))));
	} else {
		tmp = 2.0 / ((((k * k) * (t * pow(sin(k), 2.0))) / ((l * l) * cos(k))) + (2.0 * ((pow(sin(k), 2.0) * pow(t, 3.0)) / ((l * l) * cos(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 t < -2.8822792546987859e-232 or 1.48959003233934872e-174 < t

    1. Initial program 28.9

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

    2. Simplified28.9

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

    4. Applied cube-mult_binary6428.9

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

    5. Applied times-frac_binary6420.6

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

    6. Applied associate-*l*_binary6418.5

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

    7. Simplified18.5

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

    9. Applied *-un-lft-identity_binary6418.5

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

    10. Applied times-frac_binary6413.2

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

    11. Applied associate-*r*_binary6412.4

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

    12. Simplified12.4

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

    14. Applied associate-*l*_binary6411.1

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

    15. Simplified11.1

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

    17. Applied associate-*l*_binary649.7

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

    if -2.8822792546987859e-232 < t < 1.48959003233934872e-174

    1. Initial program 64.0

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

    2. Simplified64.0

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

    3. Taylor expanded around inf 42.2

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

    4. Simplified42.2

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

  4. Final simplification13.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.882279254698786 \cdot 10^{-232} \lor \neg \left(t \leq 1.4895900323393487 \cdot 10^{-174}\right):\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \ell\right) \cdot \cos k} + 2 \cdot \frac{{\sin k}^{2} \cdot {t}^{3}}{\left(\ell \cdot \ell\right) \cdot \cos k}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020231 
(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))))