Average Error: 32.5 → 9.4
Time: 13.2s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -1.6810985444739314 \cdot 10^{-15} \lor \neg \left(t \leq 5661311553777.913\right):\\ \;\;\;\;\frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right)}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \left(t \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)\right) + \frac{{\sin k}^{2} \cdot \left(k \cdot k\right)}{\ell}}{\frac{\ell}{t} \cdot \cos k}}\\ \end{array}\]
\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 -1.6810985444739314 \cdot 10^{-15} \lor \neg \left(t \leq 5661311553777.913\right):\\
\;\;\;\;\frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right)}{\frac{\ell}{t} \cdot \cos k}}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{\frac{2 \cdot \left(t \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)\right) + \frac{{\sin k}^{2} \cdot \left(k \cdot k\right)}{\ell}}{\frac{\ell}{t} \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 -1.6810985444739314e-15) (not (<= t 5661311553777.913)))
   (/
    2.0
    (/
     (* (+ 2.0 (pow (/ k t) 2.0)) (* (sin k) (* t (/ (* t (sin k)) l))))
     (* (/ l t) (cos k))))
   (/
    2.0
    (/
     (+
      (* 2.0 (* t (* (pow (sin k) 2.0) (/ t l))))
      (/ (* (pow (sin k) 2.0) (* k k)) l))
     (* (/ l t) (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 <= -1.6810985444739314e-15) || !(t <= 5661311553777.913)) {
		tmp = 2.0 / (((2.0 + pow((k / t), 2.0)) * (sin(k) * (t * ((t * sin(k)) / l)))) / ((l / t) * cos(k)));
	} else {
		tmp = 2.0 / (((2.0 * (t * (pow(sin(k), 2.0) * (t / l)))) + ((pow(sin(k), 2.0) * (k * k)) / l)) / ((l / t) * 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 < -1.6810985444739314e-15 or 5661311553777.91309 < t

    1. Initial program 22.6

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

    2. Simplified22.6

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

    4. Applied unpow3_binary6422.6

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

    5. Applied times-frac_binary6416.0

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

    6. Applied associate-*l*_binary6413.8

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

    8. Applied associate-/l*_binary647.8

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

    10. Applied tan-quot_binary647.8

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

    11. Applied associate-*l/_binary646.3

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

    12. Applied frac-times_binary643.3

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

    13. Applied associate-*l/_binary643.0

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

    14. Simplified3.0

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

    16. Applied associate-*l/_binary642.9

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

    if -1.6810985444739314e-15 < t < 5661311553777.91309

    1. Initial program 49.4

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

    2. Simplified49.4

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

    4. Applied unpow3_binary6449.4

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

    5. Applied times-frac_binary6441.5

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

    6. Applied associate-*l*_binary6440.2

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

    8. Applied associate-/l*_binary6434.1

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

    10. Applied tan-quot_binary6434.1

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

    11. Applied associate-*l/_binary6434.2

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

    12. Applied frac-times_binary6435.7

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

    13. Applied associate-*l/_binary6432.3

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

    14. Simplified32.3

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

    15. Taylor expanded around inf 21.8

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

    16. Simplified20.3

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

  4. Final simplification9.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.6810985444739314 \cdot 10^{-15} \lor \neg \left(t \leq 5661311553777.913\right):\\ \;\;\;\;\frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \left(t \cdot \frac{t \cdot \sin k}{\ell}\right)\right)}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \left(t \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)\right) + \frac{{\sin k}^{2} \cdot \left(k \cdot k\right)}{\ell}}{\frac{\ell}{t} \cdot \cos k}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020260 
(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))))