Average Error: 48.5 → 3.9
Time: 24.9s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \leq -8.037426128813863 \cdot 10^{-109} \lor \neg \left(\ell \leq 4.5304118855774756 \cdot 10^{-246}\right):\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k} \cdot \frac{2}{\frac{k}{\frac{\ell}{\frac{{\sin k}^{2}}{\cos k}}}}\\ \end{array}\]
\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}\;\ell \leq -8.037426128813863 \cdot 10^{-109} \lor \neg \left(\ell \leq 4.5304118855774756 \cdot 10^{-246}\right):\\
\;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\ell}{t}}{k} \cdot \frac{2}{\frac{k}{\frac{\ell}{\frac{{\sin k}^{2}}{\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 (<= l -8.037426128813863e-109) (not (<= l 4.5304118855774756e-246)))
   (/ 2.0 (* (/ k l) (* (/ k l) (/ (* t (pow (sin k) 2.0)) (cos k)))))
   (* (/ (/ l t) k) (/ 2.0 (/ k (/ l (/ (pow (sin k) 2.0) (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 ((l <= -8.037426128813863e-109) || !(l <= 4.5304118855774756e-246)) {
		tmp = 2.0 / ((k / l) * ((k / l) * ((t * pow(sin(k), 2.0)) / cos(k))));
	} else {
		tmp = ((l / t) / k) * (2.0 / (k / (l / (pow(sin(k), 2.0) / 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 l < -8.03742612881386299e-109 or 4.5304118855774756e-246 < l

    1. Initial program 49.6

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

    2. Simplified42.2

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

    3. Taylor expanded around inf 25.1

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

    4. Simplified25.1

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

    6. Applied associate-/l*_binary64_36124.1

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

    7. Simplified24.1

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

    9. Applied *-un-lft-identity_binary64_41424.1

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

    10. Applied times-frac_binary64_42019.3

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

    11. Applied times-frac_binary64_4207.1

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

    12. Simplified7.1

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

    13. Simplified2.3

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

    if -8.03742612881386299e-109 < l < 4.5304118855774756e-246

    1. Initial program 45.8

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

    2. Simplified37.0

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

    3. Taylor expanded around inf 17.2

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

    4. Simplified17.2

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

    6. Applied associate-/l*_binary64_36116.6

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

    7. Simplified16.6

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

    9. Applied *-un-lft-identity_binary64_41416.6

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

    10. Applied times-frac_binary64_42016.6

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

    11. Applied times-frac_binary64_4209.2

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

    12. Applied times-frac_binary64_4207.9

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

    13. Applied *-un-lft-identity_binary64_4147.9

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

    14. Applied times-frac_binary64_4207.7

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

    15. Simplified7.7

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

  4. Final simplification3.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.037426128813863 \cdot 10^{-109} \lor \neg \left(\ell \leq 4.5304118855774756 \cdot 10^{-246}\right):\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k} \cdot \frac{2}{\frac{k}{\frac{\ell}{\frac{{\sin k}^{2}}{\cos k}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020281 
(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))))