Average Error: 48.6 → 7.6
Time: 37.7s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \leq -9.149498934587888 \cdot 10^{-05}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \frac{{\left(\left|\frac{\ell}{k}\right|\right)}^{2}}{t}}{{\sin k}^{2}}\\ \mathbf{elif}\;k \leq -7.942342797387742 \cdot 10^{-100}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\ell}{{k}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{\sin k \cdot \left(t \cdot \sin k\right)}\\ \end{array}\]
\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 -9.149498934587888 \cdot 10^{-05}:\\
\;\;\;\;2 \cdot \frac{\cos k \cdot \frac{{\left(\left|\frac{\ell}{k}\right|\right)}^{2}}{t}}{{\sin k}^{2}}\\

\mathbf{elif}\;k \leq -7.942342797387742 \cdot 10^{-100}:\\
\;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\ell}{{k}^{3}}\right)\\

\mathbf{else}:\\
\;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{\sin k \cdot \left(t \cdot \sin k\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 -9.149498934587888e-05)
   (* 2.0 (/ (* (cos k) (/ (pow (fabs (/ l k)) 2.0) t)) (pow (sin k) 2.0)))
   (if (<= k -7.942342797387742e-100)
     (* 2.0 (* (/ (/ l k) t) (/ l (pow k 3.0))))
     (* 2.0 (/ (* (cos k) (* (/ l k) (/ l k))) (* (sin k) (* t (sin 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 (k <= -9.149498934587888e-05) {
		tmp = 2.0 * ((cos(k) * (pow(fabs(l / k), 2.0) / t)) / pow(sin(k), 2.0));
	} else if (k <= -7.942342797387742e-100) {
		tmp = 2.0 * (((l / k) / t) * (l / pow(k, 3.0)));
	} else {
		tmp = 2.0 * ((cos(k) * ((l / k) * (l / k))) / (sin(k) * (t * sin(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 3 regimes

  2. if k < -9.14949893458788776e-5

    1. Initial program 44.3

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

    2. Simplified36.8

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

    3. Taylor expanded around 0 19.9

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

    4. Simplified19.9

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

    6. Applied associate-/r*_binary64_36319.9

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

    7. Simplified19.9

      \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell \cdot \ell}{k \cdot k} \cdot \cos k}}{t \cdot {\sin k}^{2}}\]
    8. Using strategy rm

    9. Applied add-sqr-sqrt_binary64_44119.9

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

    10. Simplified19.9

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

    11. Simplified6.4

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

    13. Applied associate-/r*_binary64_3636.4

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

    14. Simplified6.4

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

    if -9.14949893458788776e-5 < k < -7.9423427973877422e-100

    1. Initial program 59.6

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

    2. Simplified49.3

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

    3. Taylor expanded around 0 22.8

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

    4. Simplified22.8

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

    6. Applied associate-/r*_binary64_36319.1

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

    7. Simplified19.1

      \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell \cdot \ell}{k \cdot k} \cdot \cos k}}{t \cdot {\sin k}^{2}}\]
    8. Using strategy rm

    9. Applied add-sqr-sqrt_binary64_44119.1

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

    10. Simplified19.0

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

    11. Simplified14.5

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

    12. Taylor expanded around 0 14.8

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

    13. Simplified2.6

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

    if -7.9423427973877422e-100 < k

    1. Initial program 50.0

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

    2. Simplified42.7

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

    3. Taylor expanded around 0 25.3

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

    4. Simplified25.3

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

    6. Applied associate-/r*_binary64_36323.9

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

    7. Simplified23.9

      \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell \cdot \ell}{k \cdot k} \cdot \cos k}}{t \cdot {\sin k}^{2}}\]
    8. Using strategy rm

    9. Applied add-sqr-sqrt_binary64_44123.9

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

    10. Simplified23.9

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

    11. Simplified11.6

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

    13. Applied unpow2_binary64_48411.6

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

    14. Applied associate-*r*_binary64_3599.2

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

  4. Final simplification7.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -9.149498934587888 \cdot 10^{-05}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \frac{{\left(\left|\frac{\ell}{k}\right|\right)}^{2}}{t}}{{\sin k}^{2}}\\ \mathbf{elif}\;k \leq -7.942342797387742 \cdot 10^{-100}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\ell}{{k}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{\sin k \cdot \left(t \cdot \sin k\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021091 
(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))))