Average Error: 32.4 → 11.0
Time: 28.5s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -3.400026457943122 \cdot 10^{-67}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{t}}}\\ \mathbf{elif}\;t \leq 1.859552423715104 \cdot 10^{-177}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \end{array}\]
\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 -3.400026457943122 \cdot 10^{-67}:\\
\;\;\;\;\frac{2}{\frac{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{t}}}\\

\mathbf{elif}\;t \leq 1.859552423715104 \cdot 10^{-177}:\\
\;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\ell}{t} \cdot \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\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 -3.400026457943122e-67)
   (/
    2.0
    (/
     (* (* (* (/ t l) (* t (sin k))) (tan k)) (+ 2.0 (pow (/ k t) 2.0)))
     (/ l t)))
   (if (<= t 1.859552423715104e-177)
     (*
      2.0
      (/ (* (cos k) (pow l 2.0)) (* (pow k 2.0) (* t (pow (sin k) 2.0)))))
     (*
      (/ l t)
      (/
       2.0
       (* (* (* (/ t l) (* t (sin k))) (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 <= -3.400026457943122e-67) {
		tmp = 2.0 / (((((t / l) * (t * sin(k))) * tan(k)) * (2.0 + pow((k / t), 2.0))) / (l / t));
	} else if (t <= 1.859552423715104e-177) {
		tmp = 2.0 * ((cos(k) * pow(l, 2.0)) / (pow(k, 2.0) * (t * pow(sin(k), 2.0))));
	} else {
		tmp = (l / t) * (2.0 / ((((t / l) * (t * sin(k))) * 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 < -3.40002645794312209e-67

    1. Initial program 22.5

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

    2. Simplified22.5

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

    4. Applied cube-mult_binary64_44922.5

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

    5. Applied times-frac_binary64_42516.6

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

    6. Applied associate-*l*_binary64_36014.3

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

    7. Simplified14.3

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

    9. Applied associate-/l*_binary64_3649.3

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

    11. Applied associate-*r/_binary64_3618.1

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

    12. Applied associate-*r/_binary64_3618.1

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

    13. Applied associate-*l/_binary64_3626.0

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

    14. Applied associate-*l/_binary64_3625.3

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

    if -3.40002645794312209e-67 < t < 1.85955242371510397e-177

    1. Initial program 60.1

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

    2. Simplified60.1

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

    4. Applied cube-mult_binary64_44960.1

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

    5. Applied times-frac_binary64_42553.5

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

    6. Applied associate-*l*_binary64_36053.4

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

    7. Simplified53.4

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

    9. Applied associate-/l*_binary64_36444.2

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

    10. Taylor expanded around 0 25.9

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

    if 1.85955242371510397e-177 < t

    1. Initial program 27.5

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

    2. Simplified27.5

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

    4. Applied cube-mult_binary64_44927.5

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

    5. Applied times-frac_binary64_42519.0

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

    6. Applied associate-*l*_binary64_36017.0

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

    7. Simplified17.0

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

    9. Applied associate-/l*_binary64_36411.7

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

    11. Applied associate-*r/_binary64_36111.1

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

    12. Applied associate-*r/_binary64_36111.1

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

    13. Applied associate-*l/_binary64_3629.6

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

    14. Applied associate-*l/_binary64_3628.6

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

    15. Applied associate-/r/_binary64_3658.5

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

  4. Final simplification11.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.400026457943122 \cdot 10^{-67}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{t}}}\\ \mathbf{elif}\;t \leq 1.859552423715104 \cdot 10^{-177}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021059 
(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))))