Average Error: 33.4 → 10.6
Time: 24.9s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -1.6876872750005436 \cdot 10^{-36}:\\ \;\;\;\;\frac{\frac{2}{\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 7.756230478653196 \cdot 10^{-143}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array}\]
\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}
\mathbf{if}\;t \leq -1.6876872750005436 \cdot 10^{-36}:\\
\;\;\;\;\frac{\frac{2}{\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\

\mathbf{elif}\;t \leq 7.756230478653196 \cdot 10^{-143}:\\
\;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{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
 (if (<= t -1.6876872750005436e-36)
   (/
    (/ 2.0 (* (* t (/ (* t (sin k)) l)) (* (/ t l) (tan k))))
    (+ 2.0 (pow (/ k t) 2.0)))
   (if (<= t 7.756230478653196e-143)
     (/
      2.0
      (/ (* (pow k 2.0) (* t (pow (sin k) 2.0))) (* (pow l 2.0) (cos k))))
     (/
      2.0
      (*
       (* t (/ (* t (sin k)) l))
       (* (* (/ t l) (tan k)) (+ 2.0 (pow (/ k t) 2.0))))))))
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 (t <= -1.6876872750005436e-36) {
		tmp = (2.0 / ((t * ((t * sin(k)) / l)) * ((t / l) * tan(k)))) / (2.0 + pow((k / t), 2.0));
	} else if (t <= 7.756230478653196e-143) {
		tmp = 2.0 / ((pow(k, 2.0) * (t * pow(sin(k), 2.0))) / (pow(l, 2.0) * cos(k)));
	} else {
		tmp = 2.0 / ((t * ((t * sin(k)) / l)) * (((t / l) * tan(k)) * (2.0 + pow((k / t), 2.0))));
	}
	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 < -1.6876872750005436e-36

    1. Initial program 24.1

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

    2. Simplified24.1

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

    4. Applied unpow3_binary64_48524.1

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

    5. Applied times-frac_binary64_42517.8

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

    6. Applied associate-*l*_binary64_36015.6

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

    7. Simplified15.6

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

    9. Applied associate-*r*_binary64_35915.6

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

    10. Simplified7.6

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

    12. Applied associate-*l*_binary64_3603.6

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

    14. Applied associate-/r*_binary64_3633.6

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

    if -1.6876872750005436e-36 < t < 7.75623047865319634e-143

    1. Initial program 57.5

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

    2. Simplified57.5

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

    3. Taylor expanded around 0 27.6

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

    if 7.75623047865319634e-143 < t

    1. Initial program 26.8

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

    2. Simplified26.8

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

    4. Applied unpow3_binary64_48526.8

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

    5. Applied times-frac_binary64_42518.8

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

    6. Applied associate-*l*_binary64_36016.4

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

    7. Simplified16.4

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

    9. Applied associate-*r*_binary64_35916.4

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

    10. Simplified10.6

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

    12. Applied associate-*l*_binary64_3607.5

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

    14. Applied associate-*l*_binary64_3606.2

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

    15. Simplified6.2

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.6876872750005436 \cdot 10^{-36}:\\ \;\;\;\;\frac{\frac{2}{\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \tan k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 7.756230478653196 \cdot 10^{-143}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021069 
(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))))