Average Error: 32.0 → 11.8
Time: 27.2s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
↓
\[\begin{array}{l} t_1 := t \cdot \sin k\\ t_2 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -1.3838094519414564 \cdot 10^{-52}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \left(\left(t \cdot \left(\frac{1}{\ell} \cdot t_1\right)\right) \cdot \tan k\right)\right) \cdot \left(2 + t_2\right)}\\ \mathbf{elif}\;t \leq 9.208186249841077 \cdot 10^{-142}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\cos k} + 2 \cdot \frac{{t}^{3}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot t_1\right)\right) \cdot \left(\tan k \cdot \left(-2 - t_2\right)\right)}\\ \end{array} \]
\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}
t_1 := t \cdot \sin k\\
t_2 := {\left(\frac{k}{t}\right)}^{2}\\
\mathbf{if}\;t \leq -1.3838094519414564 \cdot 10^{-52}:\\
\;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \left(\left(t \cdot \left(\frac{1}{\ell} \cdot t_1\right)\right) \cdot \tan k\right)\right) \cdot \left(2 + t_2\right)}\\

\mathbf{elif}\;t \leq 9.208186249841077 \cdot 10^{-142}:\\
\;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\cos k} + 2 \cdot \frac{{t}^{3}}{\cos k}\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot t_1\right)\right) \cdot \left(\tan k \cdot \left(-2 - t_2\right)\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
 (let* ((t_1 (* t (sin k))) (t_2 (pow (/ k t) 2.0)))
   (if (<= t -1.3838094519414564e-52)
     (/ 2.0 (* (* (/ t l) (* (* t (* (/ 1.0 l) t_1)) (tan k))) (+ 2.0 t_2)))
     (if (<= t 9.208186249841077e-142)
       (/
        2.0
        (*
         (/ (pow (sin k) 2.0) (* l l))
         (+ (/ (* t (* k k)) (cos k)) (* 2.0 (/ (pow t 3.0) (cos k))))))
       (/ -2.0 (* (* (/ t l) (* (/ t l) t_1)) (* (tan k) (- -2.0 t_2))))))))
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 * sin(k);
	double t_2 = pow((k / t), 2.0);
	double tmp;
	if (t <= -1.3838094519414564e-52) {
		tmp = 2.0 / (((t / l) * ((t * ((1.0 / l) * t_1)) * tan(k))) * (2.0 + t_2));
	} else if (t <= 9.208186249841077e-142) {
		tmp = 2.0 / ((pow(sin(k), 2.0) / (l * l)) * (((t * (k * k)) / cos(k)) + (2.0 * (pow(t, 3.0) / cos(k)))));
	} else {
		tmp = -2.0 / (((t / l) * ((t / l) * t_1)) * (tan(k) * (-2.0 - t_2)));
	}
	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 < -1.3838094519414564e-52

    1. Initial program 22.6

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

    2. Simplified22.6

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

    4. Applied cube-mult_binary6422.6

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

    5. Applied times-frac_binary6416.4

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

    6. Applied associate-*l*_binary6414.1

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

    7. Simplified14.1

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

    9. Applied *-un-lft-identity_binary6414.1

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

    10. Applied times-frac_binary648.7

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

    11. Applied associate-*r*_binary647.7

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

    12. Simplified7.7

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

    14. Applied associate-*l*_binary645.1

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

    15. Simplified5.1

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

    17. Applied div-inv_binary645.1

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

    18. Applied associate-*l*_binary645.1

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

    if -1.3838094519414564e-52 < t < 9.2081862498410771e-142

    1. Initial program 58.5

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

    2. Simplified58.5

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

    3. Taylor expanded around 0 38.2

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

    4. Simplified24.9

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

    if 9.2081862498410771e-142 < t

    1. Initial program 24.7

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

    2. Simplified24.7

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

    4. Applied cube-mult_binary6424.7

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

    5. Applied times-frac_binary6417.5

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

    6. Applied associate-*l*_binary6415.5

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

    7. Simplified15.5

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

    9. Applied *-un-lft-identity_binary6415.5

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

    10. Applied times-frac_binary6410.9

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

    11. Applied associate-*r*_binary649.9

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

    12. Simplified9.9

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

    14. Applied frac-2neg_binary649.9

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

    15. Simplified9.9

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

    16. Simplified9.9

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

  4. Final simplification11.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.3838094519414564 \cdot 10^{-52}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \left(\left(t \cdot \left(\frac{1}{\ell} \cdot \left(t \cdot \sin k\right)\right)\right) \cdot \tan k\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{elif}\;t \leq 9.208186249841077 \cdot 10^{-142}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\cos k} + 2 \cdot \frac{{t}^{3}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right)\right) \cdot \left(\tan k \cdot \left(-2 - {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021202 
(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))))