Average Error: 47.8 → 2.7
Time: 22.2s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.966348264512985 \cdot 10^{-232} \lor \neg \left(\ell \leq -4.238571491587023 \cdot 10^{-303}\right):\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \frac{\sin k \cdot \left(t \cdot \sin k\right)}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t}} \cdot \frac{k}{\frac{\ell}{\frac{{\sin k}^{2}}{\cos k}}}}\\ \end{array}\]
\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}\;\ell \leq -3.966348264512985 \cdot 10^{-232} \lor \neg \left(\ell \leq -4.238571491587023 \cdot 10^{-303}\right):\\
\;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \frac{\sin k \cdot \left(t \cdot \sin k\right)}{\cos k}\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t}} \cdot \frac{k}{\frac{\ell}{\frac{{\sin k}^{2}}{\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 (<= l -3.966348264512985e-232) (not (<= l -4.238571491587023e-303)))
   (/ 2.0 (* (/ k l) (* (/ k l) (/ (* (sin k) (* t (sin k))) (cos k)))))
   (/ 2.0 (* (/ k (/ l t)) (/ k (/ l (/ (pow (sin k) 2.0) (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 ((l <= -3.966348264512985e-232) || !(l <= -4.238571491587023e-303)) {
		tmp = 2.0 / ((k / l) * ((k / l) * ((sin(k) * (t * sin(k))) / cos(k))));
	} else {
		tmp = 2.0 / ((k / (l / t)) * (k / (l / (pow(sin(k), 2.0) / 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 l < -3.96634826451298485e-232 or -4.2385714915870229e-303 < l

    1. Initial program 48.0

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

    2. Simplified40.3

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

    3. Taylor expanded around 0 23.0

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

    4. Simplified23.0

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

    6. Applied associate-/l*_binary6422.3

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

    7. Simplified22.3

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

    9. Applied *-un-lft-identity_binary6422.3

      \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\ell \cdot \ell}{\color{blue}{1 \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}}}}\]

    10. Applied times-frac_binary6417.7

      \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\ell}{1} \cdot \frac{\ell}{\frac{t \cdot {\sin k}^{2}}{\cos k}}}}}\]

    11. Applied times-frac_binary647.0

      \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{\ell}{1}} \cdot \frac{k}{\frac{\ell}{\frac{t \cdot {\sin k}^{2}}{\cos k}}}}}\]

    12. Simplified7.0

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

    13. Simplified3.5

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

    15. Applied unpow2_binary643.5

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

    16. Applied associate-*r*_binary642.2

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

    if -3.96634826451298485e-232 < l < -4.2385714915870229e-303

    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. Simplified34.9

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

    3. Taylor expanded around 0 17.8

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

    4. Simplified17.8

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

    6. Applied associate-/l*_binary6417.6

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

    7. Simplified17.6

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

    9. Applied *-un-lft-identity_binary6417.6

      \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\ell \cdot \ell}{\frac{t \cdot {\sin k}^{2}}{\color{blue}{1 \cdot \cos k}}}}}\]

    10. Applied times-frac_binary6417.6

      \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\ell \cdot \ell}{\color{blue}{\frac{t}{1} \cdot \frac{{\sin k}^{2}}{\cos k}}}}}\]

    11. Applied times-frac_binary6410.3

      \[\leadsto \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\ell}{\frac{t}{1}} \cdot \frac{\ell}{\frac{{\sin k}^{2}}{\cos k}}}}}\]

    12. Applied times-frac_binary647.9

      \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{\ell}{\frac{t}{1}}} \cdot \frac{k}{\frac{\ell}{\frac{{\sin k}^{2}}{\cos k}}}}}\]

    13. Simplified7.9

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

  4. Final simplification2.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.966348264512985 \cdot 10^{-232} \lor \neg \left(\ell \leq -4.238571491587023 \cdot 10^{-303}\right):\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \frac{\sin k \cdot \left(t \cdot \sin k\right)}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t}} \cdot \frac{k}{\frac{\ell}{\frac{{\sin k}^{2}}{\cos k}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021126 
(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))))