Average Error: 33.1 → 11.2
Time: 14.3s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \leq -2.2065459294652522 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{\cos k} \cdot \left(2 \cdot \left(t \cdot \frac{t}{\ell}\right) + \frac{k \cdot k}{\ell}\right)}{\frac{\ell}{t}}}\\ \mathbf{elif}\;k \leq 2.4797750158614884 \cdot 10^{-177}:\\ \;\;\;\;\frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(t \cdot \frac{\sin k \cdot t}{\ell}\right)\right)}{\frac{\ell}{t}}}\\ \mathbf{elif}\;k \leq 4.443876121186734 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{\cos k} \cdot \left(2 \cdot \left(t \cdot \frac{t}{\ell}\right) + \frac{k \cdot k}{\ell}\right)}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell} + 2 \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \end{array}\]
\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}\;k \leq -2.2065459294652522 \cdot 10^{-94}:\\
\;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{\cos k} \cdot \left(2 \cdot \left(t \cdot \frac{t}{\ell}\right) + \frac{k \cdot k}{\ell}\right)}{\frac{\ell}{t}}}\\

\mathbf{elif}\;k \leq 2.4797750158614884 \cdot 10^{-177}:\\
\;\;\;\;\frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(t \cdot \frac{\sin k \cdot t}{\ell}\right)\right)}{\frac{\ell}{t}}}\\

\mathbf{elif}\;k \leq 4.443876121186734 \cdot 10^{+154}:\\
\;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{\cos k} \cdot \left(2 \cdot \left(t \cdot \frac{t}{\ell}\right) + \frac{k \cdot k}{\ell}\right)}{\frac{\ell}{t}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell} + 2 \cdot \frac{{t}^{3}}{\ell \cdot \ell}\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 (<= k -2.2065459294652522e-94)
   (/
    2.0
    (/
     (* (/ (pow (sin k) 2.0) (cos k)) (+ (* 2.0 (* t (/ t l))) (/ (* k k) l)))
     (/ l t)))
   (if (<= k 2.4797750158614884e-177)
     (/
      2.0
      (/
       (* (+ 2.0 (pow (/ k t) 2.0)) (* (tan k) (* t (/ (* (sin k) t) l))))
       (/ l t)))
     (if (<= k 4.443876121186734e+154)
       (/
        2.0
        (/
         (*
          (/ (pow (sin k) 2.0) (cos k))
          (+ (* 2.0 (* t (/ t l))) (/ (* k k) l)))
         (/ l t)))
       (/
        2.0
        (*
         (/ (pow (sin k) 2.0) (cos k))
         (+ (/ (* k (* k t)) (* l l)) (* 2.0 (/ (pow t 3.0) (* l l))))))))))
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 (k <= -2.2065459294652522e-94) {
		tmp = 2.0 / (((pow(sin(k), 2.0) / cos(k)) * ((2.0 * (t * (t / l))) + ((k * k) / l))) / (l / t));
	} else if (k <= 2.4797750158614884e-177) {
		tmp = 2.0 / (((2.0 + pow((k / t), 2.0)) * (tan(k) * (t * ((sin(k) * t) / l)))) / (l / t));
	} else if (k <= 4.443876121186734e+154) {
		tmp = 2.0 / (((pow(sin(k), 2.0) / cos(k)) * ((2.0 * (t * (t / l))) + ((k * k) / l))) / (l / t));
	} else {
		tmp = 2.0 / ((pow(sin(k), 2.0) / cos(k)) * (((k * (k * t)) / (l * l)) + (2.0 * (pow(t, 3.0) / (l * l)))));
	}
	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 3 regimes

  2. if k < -2.20654592946525221e-94 or 2.47977501586148838e-177 < k < 4.44387612118673385e154

    1. Initial program 31.7

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

    2. Simplified31.7

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

    4. Applied unpow3_binary6431.7

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

    5. Applied times-frac_binary6423.9

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

    6. Applied associate-*l*_binary6423.1

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

    8. Applied associate-/l*_binary6416.8

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

    10. Applied associate-*l/_binary6416.8

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

    11. Applied associate-*l/_binary6416.0

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

    12. Applied associate-*l/_binary6414.2

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

    13. Simplified14.2

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

    14. Taylor expanded around inf 13.3

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

    15. Simplified9.0

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

    if -2.20654592946525221e-94 < k < 2.47977501586148838e-177

    1. Initial program 38.4

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

    2. Simplified38.4

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

    4. Applied unpow3_binary6438.4

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

    5. Applied times-frac_binary6432.9

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

    6. Applied associate-*l*_binary6424.2

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

    8. Applied associate-/l*_binary6417.8

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

    10. Applied associate-*l/_binary6413.1

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

    11. Applied associate-*l/_binary649.3

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

    12. Applied associate-*l/_binary649.3

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

    13. Simplified9.3

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

    15. Applied associate-*l/_binary648.6

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

    if 4.44387612118673385e154 < k

    1. Initial program 32.9

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

    2. Simplified32.9

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

    4. Applied unpow3_binary6432.9

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

    5. Applied times-frac_binary6426.5

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

    6. Applied associate-*l*_binary6426.5

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

    8. Applied associate-/l*_binary6421.3

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

    10. Applied associate-*l/_binary6421.3

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

    11. Applied associate-*l/_binary6421.3

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

    12. Applied associate-*l/_binary6419.0

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

    13. Simplified19.0

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

    14. Taylor expanded around inf 27.7

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

    15. Simplified22.5

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

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2.2065459294652522 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{\cos k} \cdot \left(2 \cdot \left(t \cdot \frac{t}{\ell}\right) + \frac{k \cdot k}{\ell}\right)}{\frac{\ell}{t}}}\\ \mathbf{elif}\;k \leq 2.4797750158614884 \cdot 10^{-177}:\\ \;\;\;\;\frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(t \cdot \frac{\sin k \cdot t}{\ell}\right)\right)}{\frac{\ell}{t}}}\\ \mathbf{elif}\;k \leq 4.443876121186734 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{\cos k} \cdot \left(2 \cdot \left(t \cdot \frac{t}{\ell}\right) + \frac{k \cdot k}{\ell}\right)}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell} + 2 \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020274 
(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))))