Average Error: 48.0 → 7.8
Time: 28.1s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
↓
\[\begin{array}{l} t_1 := t \cdot {\sin k}^{2}\\ \mathbf{if}\;\ell \leq -2.1513562525415348 \cdot 10^{+127} \lor \neg \left(\ell \leq -6.612126816598227 \cdot 10^{+43}\right) \land \left(\ell \leq 2.2868762441407734 \cdot 10^{-51} \lor \neg \left(\ell \leq 7.006789343459675 \cdot 10^{+148}\right)\right):\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k} \cdot \frac{\ell}{\frac{k}{\frac{\ell}{t_1}}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot {\ell}^{2}}{k}}{k \cdot t_1}\\ \end{array} \]
\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}
t_1 := t \cdot {\sin k}^{2}\\
\mathbf{if}\;\ell \leq -2.1513562525415348 \cdot 10^{+127} \lor \neg \left(\ell \leq -6.612126816598227 \cdot 10^{+43}\right) \land \left(\ell \leq 2.2868762441407734 \cdot 10^{-51} \lor \neg \left(\ell \leq 7.006789343459675 \cdot 10^{+148}\right)\right):\\
\;\;\;\;2 \cdot \left(\frac{\cos k}{k} \cdot \frac{\ell}{\frac{k}{\frac{\ell}{t_1}}}\right)\\

\mathbf{else}:\\
\;\;\;\;2 \cdot \frac{\frac{\cos k \cdot {\ell}^{2}}{k}}{k \cdot t_1}\\


\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
 (let* ((t_1 (* t (pow (sin k) 2.0))))
   (if (or (<= l -2.1513562525415348e+127)
           (and (not (<= l -6.612126816598227e+43))
                (or (<= l 2.2868762441407734e-51)
                    (not (<= l 7.006789343459675e+148)))))
     (* 2.0 (* (/ (cos k) k) (/ l (/ k (/ l t_1)))))
     (* 2.0 (/ (/ (* (cos k) (pow l 2.0)) k) (* k t_1))))))
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 t_1 = t * pow(sin(k), 2.0);
	double tmp;
	if ((l <= -2.1513562525415348e+127) || (!(l <= -6.612126816598227e+43) && ((l <= 2.2868762441407734e-51) || !(l <= 7.006789343459675e+148)))) {
		tmp = 2.0 * ((cos(k) / k) * (l / (k / (l / t_1))));
	} else {
		tmp = 2.0 * (((cos(k) * pow(l, 2.0)) / k) / (k * t_1));
	}
	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 < -2.1513562525415348e127 or -6.6121268165982273e43 < l < 2.28687624414077338e-51 or 7.0067893434596753e148 < l

    1. Initial program 48.2

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

    2. Simplified41.0

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

    3. Taylor expanded in t around 0 24.6

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

    4. Applied unpow2_binary6424.6

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

    5. Applied associate-*l*_binary6423.6

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

    6. Applied times-frac_binary6422.6

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

    7. Applied unpow2_binary6422.6

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

    8. Applied associate-/l*_binary6414.2

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

    9. Simplified9.1

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

    if -2.1513562525415348e127 < l < -6.6121268165982273e43 or 2.28687624414077338e-51 < l < 7.0067893434596753e148

    1. Initial program 47.2

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

    2. Simplified38.9

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

    3. Taylor expanded in t around 0 17.9

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

    4. Applied unpow2_binary6417.9

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

    5. Applied associate-*l*_binary6411.5

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

    6. Applied associate-/r*_binary643.0

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

  4. Final simplification7.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.1513562525415348 \cdot 10^{+127} \lor \neg \left(\ell \leq -6.612126816598227 \cdot 10^{+43}\right) \land \left(\ell \leq 2.2868762441407734 \cdot 10^{-51} \lor \neg \left(\ell \leq 7.006789343459675 \cdot 10^{+148}\right)\right):\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k} \cdot \frac{\ell}{\frac{k}{\frac{\ell}{t \cdot {\sin k}^{2}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot {\ell}^{2}}{k}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(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))))