Average Error: 31.6 → 11.6
Time: 17.3s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
↓
\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -9.529228915736491 \cdot 10^{-157}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot t_1\right)}\\ \mathbf{elif}\;t \leq 3.889113154566088 \cdot 10^{-138}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \left(\sqrt{t} \cdot \left(\sin k \cdot \frac{\sqrt{t}}{\ell}\right)\right)\right)\right)\right)}\\ \end{array} \]
\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 := 2 + {\left(\frac{k}{t}\right)}^{2}\\
\mathbf{if}\;t \leq -9.529228915736491 \cdot 10^{-157}:\\
\;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot t_1\right)}\\

\mathbf{elif}\;t \leq 3.889113154566088 \cdot 10^{-138}:\\
\;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{t_1 \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \left(\sqrt{t} \cdot \left(\sin k \cdot \frac{\sqrt{t}}{\ell}\right)\right)\right)\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 (+ 2.0 (pow (/ k t) 2.0))))
   (if (<= t -9.529228915736491e-157)
     (/ 2.0 (* (/ t l) (* (* (* t (/ (* t (sin k)) l)) (tan k)) t_1)))
     (if (<= t 3.889113154566088e-138)
       (/
        2.0
        (/ (* (pow k 2.0) (* t (pow (sin k) 2.0))) (* (cos k) (pow l 2.0))))
       (/
        2.0
        (*
         t_1
         (*
          (tan k)
          (* (/ t l) (* t (* (sqrt t) (* (sin k) (/ (sqrt t) l))))))))))))
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 = 2.0 + pow((k / t), 2.0);
	double tmp;
	if (t <= -9.529228915736491e-157) {
		tmp = 2.0 / ((t / l) * (((t * ((t * sin(k)) / l)) * tan(k)) * t_1));
	} else if (t <= 3.889113154566088e-138) {
		tmp = 2.0 / ((pow(k, 2.0) * (t * pow(sin(k), 2.0))) / (cos(k) * pow(l, 2.0)));
	} else {
		tmp = 2.0 / (t_1 * (tan(k) * ((t / l) * (t * (sqrt(t) * (sin(k) * (sqrt(t) / l)))))));
	}
	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 < -9.52922891573649142e-157

    1. Initial program 26.0

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

    2. Simplified26.0

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

    3. Applied cube-mult_binary6426.0

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

    4. Applied times-frac_binary6417.7

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

    5. Applied associate-*l*_binary6415.6

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

    6. Applied *-un-lft-identity_binary6415.6

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

    7. Applied times-frac_binary6411.4

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

    8. Applied associate-*l*_binary6410.4

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

    9. Applied associate-*l*_binary648.8

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

    10. Simplified8.8

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

    11. Applied associate-*l*_binary647.7

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

    if -9.52922891573649142e-157 < t < 3.889113154566088e-138

    1. Initial program 64.0

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

    2. Simplified64.0

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

    3. Taylor expanded in t around 0 26.8

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

    if 3.889113154566088e-138 < t

    1. Initial program 23.9

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

    2. Simplified23.9

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

    3. Applied cube-mult_binary6423.9

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

    4. Applied times-frac_binary6416.7

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

    5. Applied associate-*l*_binary6414.4

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

    6. Applied *-un-lft-identity_binary6414.4

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

    7. Applied times-frac_binary649.7

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

    8. Applied associate-*l*_binary648.8

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

    9. Applied *-un-lft-identity_binary648.8

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

    10. Applied add-sqr-sqrt_binary648.8

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

    11. Applied times-frac_binary648.8

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

    12. Applied associate-*l*_binary649.3

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

    13. Simplified9.3

      \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\frac{t}{1} \cdot \left(\frac{\sqrt{t}}{1} \cdot \color{blue}{\left(\sin k \cdot \frac{\sqrt{t}}{\ell}\right)}\right)\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification11.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.529228915736491 \cdot 10^{-157}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\left(\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq 3.889113154566088 \cdot 10^{-138}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \left(\sqrt{t} \cdot \left(\sin k \cdot \frac{\sqrt{t}}{\ell}\right)\right)\right)\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021340 
(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))))