Average Error: 48.1 → 13.8
Time: 22.4s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \leq 4.842177147515977 \cdot 10^{-74}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{k}{\ell}} \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;k \leq 1.0962257049040088 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{0.16666666666666666 \cdot \frac{{k}^{6}}{\frac{\ell}{\frac{t}{\ell}}} + \frac{{k}^{4}}{\frac{\ell}{\frac{t}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{\sqrt{k}} \cdot \frac{\ell}{\sqrt{k}}\right)}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array}\]
\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 4.842177147515977 \cdot 10^{-74}:\\
\;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{k}{\ell}} \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\\

\mathbf{elif}\;k \leq 1.0962257049040088 \cdot 10^{-13}:\\
\;\;\;\;\frac{2}{0.16666666666666666 \cdot \frac{{k}^{6}}{\frac{\ell}{\frac{t}{\ell}}} + \frac{{k}^{4}}{\frac{\ell}{\frac{t}{\ell}}}}\\

\mathbf{else}:\\
\;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{\sqrt{k}} \cdot \frac{\ell}{\sqrt{k}}\right)}{k \cdot \left(t \cdot {\sin k}^{2}\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 4.842177147515977e-74)
   (* 2.0 (/ (* (/ l (/ k l)) (cos k)) (* k (* t (pow (sin k) 2.0)))))
   (if (<= k 1.0962257049040088e-13)
     (/
      2.0
      (+
       (* 0.16666666666666666 (/ (pow k 6.0) (/ l (/ t l))))
       (/ (pow k 4.0) (/ l (/ t l)))))
     (*
      2.0
      (/
       (* (cos k) (* (/ l (sqrt k)) (/ l (sqrt k))))
       (* k (* t (pow (sin k) 2.0))))))))
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 (k <= 4.842177147515977e-74) {
		tmp = 2.0 * (((l / (k / l)) * cos(k)) / (k * (t * pow(sin(k), 2.0))));
	} else if (k <= 1.0962257049040088e-13) {
		tmp = 2.0 / ((0.16666666666666666 * (pow(k, 6.0) / (l / (t / l)))) + (pow(k, 4.0) / (l / (t / l))));
	} else {
		tmp = 2.0 * ((cos(k) * ((l / sqrt(k)) * (l / sqrt(k)))) / (k * (t * pow(sin(k), 2.0))));
	}
	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 k < 4.8421771475159773e-74

    1. Initial program 49.5

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

    2. Simplified42.5

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

    3. Taylor expanded around inf 26.4

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

    4. Simplified26.4

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

    6. Applied associate-*l*_binary6424.6

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

    8. Applied associate-/r*_binary6422.1

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

    9. Simplified22.1

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

    11. Applied associate-/l*_binary6417.1

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

    if 4.8421771475159773e-74 < k < 1.0962257049040088e-13

    1. Initial program 60.4

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

    2. Simplified49.8

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

    3. Taylor expanded around 0 24.0

      \[\leadsto \frac{2}{\color{blue}{0.16666666666666666 \cdot \frac{{k}^{6} \cdot t}{{\ell}^{2}} + \frac{{k}^{4} \cdot t}{{\ell}^{2}}}}\]

    4. Simplified15.1

      \[\leadsto \frac{2}{\color{blue}{0.16666666666666666 \cdot \frac{{k}^{6}}{\frac{\ell}{\frac{t}{\ell}}} + \frac{{k}^{4}}{\frac{\ell}{\frac{t}{\ell}}}}}\]

    if 1.0962257049040088e-13 < k

    1. Initial program 44.5

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

    2. Simplified36.4

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

    3. Taylor expanded around inf 18.8

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

    4. Simplified18.8

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

    6. Applied associate-*l*_binary6416.1

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

    8. Applied associate-/r*_binary6414.0

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

    9. Simplified14.0

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

    11. Applied add-sqr-sqrt_binary6414.1

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

    12. Applied times-frac_binary649.0

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

  4. Final simplification13.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.842177147515977 \cdot 10^{-74}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{k}{\ell}} \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;k \leq 1.0962257049040088 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{0.16666666666666666 \cdot \frac{{k}^{6}}{\frac{\ell}{\frac{t}{\ell}}} + \frac{{k}^{4}}{\frac{\ell}{\frac{t}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{\sqrt{k}} \cdot \frac{\ell}{\sqrt{k}}\right)}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020232 
(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))))