Average Error: 32.7 → 10.6
Time: 12.2s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -4.953282748199605 \cdot 10^{-30} \lor \neg \left(t \leq 4.3543219557651655 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{1}{\frac{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\frac{\frac{\ell}{t}}{\tan k}}} \cdot \frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\cos k} + 2 \cdot \frac{{t}^{3}}{\cos k}\right)}\\ \end{array}\]
\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}\;t \leq -4.953282748199605 \cdot 10^{-30} \lor \neg \left(t \leq 4.3543219557651655 \cdot 10^{-32}\right):\\
\;\;\;\;\frac{1}{\frac{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\frac{\frac{\ell}{t}}{\tan k}}} \cdot \frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\cos k} + 2 \cdot \frac{{t}^{3}}{\cos 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 (<= t -4.953282748199605e-30) (not (<= t 4.3543219557651655e-32)))
   (*
    (/ 1.0 (/ (* t (* (/ t l) (sin k))) (/ (/ l t) (tan k))))
    (/ 2.0 (+ 2.0 (pow (/ k t) 2.0))))
   (/
    2.0
    (*
     (/ (pow (sin k) 2.0) (* l l))
     (+ (/ (* t (* k k)) (cos k)) (* 2.0 (/ (pow t 3.0) (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 ((t <= -4.953282748199605e-30) || !(t <= 4.3543219557651655e-32)) {
		tmp = (1.0 / ((t * ((t / l) * sin(k))) / ((l / t) / tan(k)))) * (2.0 / (2.0 + pow((k / t), 2.0)));
	} else {
		tmp = 2.0 / ((pow(sin(k), 2.0) / (l * l)) * (((t * (k * k)) / cos(k)) + (2.0 * (pow(t, 3.0) / 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 t < -4.9532827481996048e-30 or 4.35432195576516549e-32 < t

    1. Initial program 23.0

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

    2. Simplified23.0

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

    4. Applied cube-mult_binary6423.0

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

    5. Applied times-frac_binary6416.6

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

    6. Applied associate-*l*_binary6414.1

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

    8. Applied *-un-lft-identity_binary6414.1

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

    9. Applied times-frac_binary648.4

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

    10. Applied associate-*l*_binary647.1

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

    12. Applied *-un-lft-identity_binary647.1

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

    13. Applied times-frac_binary647.0

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

    14. Simplified4.2

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

    16. Applied clear-num_binary644.4

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

    17. Simplified3.6

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

    if -4.9532827481996048e-30 < t < 4.35432195576516549e-32

    1. Initial program 52.5

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

    2. Simplified52.5

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

    3. Taylor expanded around inf 35.6

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

    4. Simplified24.8

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.953282748199605 \cdot 10^{-30} \lor \neg \left(t \leq 4.3543219557651655 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{1}{\frac{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\frac{\frac{\ell}{t}}{\tan k}}} \cdot \frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\cos k} + 2 \cdot \frac{{t}^{3}}{\cos k}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020233 
(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))))