Average Error: 48.2 → 29.9
Time: 30.9s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.289662399796905 \cdot 10^{-58} \lor \neg \left(t \le 1.45077355592580413 \cdot 10^{-133}\right):\\ \;\;\;\;\frac{1}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{2 \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)} \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{\frac{2 \cdot 1}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)}}\right)}^{\left(\frac{\ell \cdot \ell}{\sin k}\right)}\right)\\ \end{array}\]
\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 \le -1.289662399796905 \cdot 10^{-58} \lor \neg \left(t \le 1.45077355592580413 \cdot 10^{-133}\right):\\
\;\;\;\;\frac{1}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{2 \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)} \cdot \frac{\ell}{\sin k}\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left({\left(e^{\frac{2 \cdot 1}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)}}\right)}^{\left(\frac{\ell \cdot \ell}{\sin k}\right)}\right)\\

\end{array}
double code(double t, double l, double k) {
	return ((double) (2.0 / ((double) (((double) (((double) (((double) (((double) pow(t, 3.0)) / ((double) (l * l)))) * ((double) sin(k)))) * ((double) tan(k)))) * ((double) (((double) (1.0 + ((double) pow(((double) (k / t)), 2.0)))) - 1.0))))));
}
↓
double code(double t, double l, double k) {
	double VAR;
	if (((t <= -1.289662399796905e-58) || !(t <= 1.4507735559258041e-133))) {
		VAR = ((double) (((double) (1.0 / ((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))))) * ((double) (((double) (((double) (2.0 * l)) / ((double) (((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))) * ((double) (((double) pow(t, 3.0)) * ((double) tan(k)))))))) * ((double) (l / ((double) sin(k))))))));
	} else {
		VAR = ((double) log(((double) pow(((double) exp(((double) (((double) (2.0 * 1.0)) / ((double) (((double) pow(((double) (k / t)), ((double) (2.0 * ((double) (2.0 / 2.0)))))) * ((double) (((double) pow(t, 3.0)) * ((double) tan(k)))))))))), ((double) (((double) (l * l)) / ((double) sin(k))))))));
	}
	return VAR;
}

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 < -1.289662399796905e-58 or 1.45077355592580413e-133 < t

    1. Initial program 43.6

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

    2. Simplified32.7

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

    4. Applied sqr-pow32.7

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

    5. Applied associate-*l*28.1

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

    7. Applied times-frac27.8

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

    9. Applied *-un-lft-identity27.8

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

    10. Applied times-frac27.8

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

    11. Applied associate-*l*27.1

      \[\leadsto \color{blue}{\frac{1}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{2}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)} \cdot \frac{\ell \cdot \ell}{\sin k}\right)}\]
    12. Using strategy rm

    13. Applied *-un-lft-identity27.1

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

    14. Applied times-frac26.4

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

    15. Applied associate-*r*22.7

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

    16. Simplified22.6

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

    if -1.289662399796905e-58 < t < 1.45077355592580413e-133

    1. Initial program 59.5

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

    2. Simplified59.0

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

    4. Applied sqr-pow59.0

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

    5. Applied associate-*l*58.1

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

    7. Applied times-frac58.1

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

    9. Applied add-log-exp60.2

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

    10. Simplified47.8

      \[\leadsto \log \color{blue}{\left({\left(e^{\frac{2 \cdot 1}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)}}\right)}^{\left(\frac{\ell \cdot \ell}{\sin k}\right)}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification29.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.289662399796905 \cdot 10^{-58} \lor \neg \left(t \le 1.45077355592580413 \cdot 10^{-133}\right):\\ \;\;\;\;\frac{1}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{2 \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)} \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{\frac{2 \cdot 1}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)}}\right)}^{\left(\frac{\ell \cdot \ell}{\sin k}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020173 
(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))))