Average Error: 48.0 → 27.5
Time: 25.3s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -1.8953370375107 \cdot 10^{-192}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq 2.0914496211353583 \cdot 10^{-88}:\\ \;\;\;\;0\\ \mathbf{elif}\;t \leq 1.603545270302865 \cdot 10^{+81}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{k}{t}\right)}^{2} \cdot \frac{{t}^{3}}{\frac{\ell}{\sin k \cdot \sin k}}}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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 -1.8953370375107 \cdot 10^{-192}:\\
\;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\

\mathbf{elif}\;t \leq 2.0914496211353583 \cdot 10^{-88}:\\
\;\;\;\;0\\

\mathbf{elif}\;t \leq 1.603545270302865 \cdot 10^{+81}:\\
\;\;\;\;\frac{2}{\frac{{\left(\frac{k}{t}\right)}^{2} \cdot \frac{{t}^{3}}{\frac{\ell}{\sin k \cdot \sin k}}}{\ell \cdot \cos k}}\\

\mathbf{else}:\\
\;\;\;\;0\\

\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 (<= t -1.8953370375107e-192)
   (/
    2.0
    (* (/ t l) (* (* (* t (/ t l)) (sin k)) (* (tan k) (pow (/ k t) 2.0)))))
   (if (<= t 2.0914496211353583e-88)
     0.0
     (if (<= t 1.603545270302865e+81)
       (/
        2.0
        (/
         (* (pow (/ k t) 2.0) (/ (pow t 3.0) (/ l (* (sin k) (sin k)))))
         (* l (cos k))))
       0.0))))
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 <= -1.8953370375107e-192) {
		tmp = 2.0 / ((t / l) * (((t * (t / l)) * sin(k)) * (tan(k) * pow((k / t), 2.0))));
	} else if (t <= 2.0914496211353583e-88) {
		tmp = 0.0;
	} else if (t <= 1.603545270302865e+81) {
		tmp = 2.0 / ((pow((k / t), 2.0) * (pow(t, 3.0) / (l / (sin(k) * sin(k))))) / (l * cos(k)));
	} else {
		tmp = 0.0;
	}
	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 t < -1.8953370375107e-192

    1. Initial program 45.3

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

    2. Simplified35.7

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

    4. Applied cube-mult_binary6435.7

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

    5. Applied times-frac_binary6428.6

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

    6. Simplified25.4

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

    8. Applied associate-*l*_binary6425.5

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

    10. Applied associate-*l*_binary6425.3

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

    11. Simplified25.3

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

    13. Applied associate-*l*_binary6422.9

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

    if -1.8953370375107e-192 < t < 2.0914496211353583e-88 or 1.6035452703028649e81 < t

    1. Initial program 56.4

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

    2. Simplified50.0

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

    4. Applied cube-mult_binary6450.0

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

    5. Applied times-frac_binary6444.7

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

    6. Simplified38.5

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

    7. Taylor expanded around 0 35.9

      \[\leadsto \color{blue}{0}\]

    if 2.0914496211353583e-88 < t < 1.6035452703028649e81

    1. Initial program 29.3

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

    2. Simplified22.0

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

    4. Applied cube-mult_binary6422.0

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

    5. Applied times-frac_binary6419.2

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

    6. Simplified19.2

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

    8. Applied tan-quot_binary6419.2

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

    9. Applied associate-*l/_binary6419.2

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

    10. Applied associate-*l/_binary6417.6

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

    11. Applied frac-times_binary6416.9

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

    12. Applied associate-*l/_binary6415.7

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

    13. Simplified14.9

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

  4. Final simplification27.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8953370375107 \cdot 10^{-192}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq 2.0914496211353583 \cdot 10^{-88}:\\ \;\;\;\;0\\ \mathbf{elif}\;t \leq 1.603545270302865 \cdot 10^{+81}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{k}{t}\right)}^{2} \cdot \frac{{t}^{3}}{\frac{\ell}{\sin k \cdot \sin k}}}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2020277 
(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))))