Average Error: 48.3 → 7.6
Time: 29.2s
Precision: binary64
Cost: 20552
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.9288477212394867 \cdot 10^{+98} \lor \neg \left(\ell \leq 5.655539911333164 \cdot 10^{+206}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\right)\right)\\ \end{array}\]
\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}
\mathbf{if}\;\ell \leq -1.9288477212394867 \cdot 10^{+98} \lor \neg \left(\ell \leq 5.655539911333164 \cdot 10^{+206}\right):\\
\;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\

\mathbf{else}:\\
\;\;\;\;2 \cdot \left(\ell \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{k \cdot \left(t \cdot {\sin k}^{2}\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
 (if (or (<= l -1.9288477212394867e+98) (not (<= l 5.655539911333164e+206)))
   (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* t (pow (sin k) 2.0)))))
   (* 2.0 (* l (* (/ l k) (/ (cos k) (* k (* t (pow (sin k) 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 ((l <= -1.9288477212394867e+98) || !(l <= 5.655539911333164e+206)) {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * pow(sin(k), 2.0))));
	} else {
		tmp = 2.0 * (l * ((l / k) * (cos(k) / (k * (t * pow(sin(k), 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

Alternatives

Alternative 1
Error6.8
Cost40129
\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 3.603875151524802 \cdot 10^{+196}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{\sqrt[3]{k} \cdot \sqrt[3]{k}} \cdot \left(\frac{\cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \frac{\ell}{\sqrt[3]{k}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array}\]
Alternative 2
Error7.6
Cost20808
\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 1.1539497427612818 \cdot 10^{-248} \lor \neg \left(\ell \cdot \ell \leq 4.081322563393444 \cdot 10^{+196}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \left(\ell \cdot \frac{\ell}{k}\right)\right)\\ \end{array}\]
Alternative 3
Error12.6
Cost20552
\[\begin{array}{l} \mathbf{if}\;k \leq -3.6547036640213393 \cdot 10^{-07} \lor \neg \left(k \leq -6.122936130232856 \cdot 10^{-54}\right):\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \left(\ell \cdot \frac{\ell}{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\frac{\ell}{\frac{t}{\ell}}}}\\ \end{array}\]
Alternative 4
Error12.5
Cost20552
\[\begin{array}{l} \mathbf{if}\;k \leq -1.1852346881225905 \cdot 10^{-05} \lor \neg \left(k \leq -7.063681446027815 \cdot 10^{-56}\right):\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \frac{\ell}{\frac{k}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\frac{\ell}{\frac{t}{\ell}}}}\\ \end{array}\]
Alternative 5
Error17.5
Cost20552
\[\begin{array}{l} \mathbf{if}\;k \leq -9.151404920080495 \cdot 10^{-06} \lor \neg \left(k \leq -1.3436020588189278 \cdot 10^{-52}\right):\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{k}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\frac{\ell}{\frac{t}{\ell}}}}\\ \end{array}\]
Alternative 6
Error28.8
Cost14786
\[\begin{array}{l} \mathbf{if}\;t \leq -8.783101688769685 \cdot 10^{+89}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{k} \cdot \frac{\cos k}{k \cdot \left(t \cdot \left(k \cdot k\right)\right)}\right)\\ \mathbf{elif}\;t \leq 2.167343341657803 \cdot 10^{-157}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\frac{\ell}{\frac{t}{\ell}}} + 0.16666666666666666 \cdot \frac{{k}^{6}}{\frac{\ell}{\frac{t}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\\ \end{array}\]
Alternative 7
Error28.7
Cost8130
\[\begin{array}{l} \mathbf{if}\;t \leq -3.575766571680626 \cdot 10^{-125}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{k} \cdot \frac{\cos k}{k \cdot \left(t \cdot \left(k \cdot k\right)\right)}\right)\\ \mathbf{elif}\;t \leq 9.479879216596787 \cdot 10^{-157}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\frac{\ell}{\frac{t}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\\ \end{array}\]
Alternative 8
Error28.3
Cost7816
\[\begin{array}{l} \mathbf{if}\;t \leq -2.4575930907319535 \cdot 10^{-127} \lor \neg \left(t \leq 7.777500190582001 \cdot 10^{-156}\right):\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{k} \cdot \frac{\cos k}{k \cdot \left(t \cdot \left(k \cdot k\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\frac{\ell}{\frac{t}{\ell}}}}\\ \end{array}\]
Alternative 9
Error28.8
Cost7624
\[\begin{array}{l} \mathbf{if}\;k \leq -1.3852239520341363 \cdot 10^{-58} \lor \neg \left(k \leq 3.1551806453450644 \cdot 10^{-78}\right):\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\frac{\ell}{\frac{t}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{k} \cdot \frac{1}{t \cdot {k}^{3}}\right)\\ \end{array}\]
Alternative 10
Error29.9
Cost7040
\[\frac{2}{\frac{{k}^{4}}{\frac{\ell}{\frac{t}{\ell}}}}\]
Alternative 11
Error31.4
Cost7040
\[2 \cdot \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\]
Alternative 12
Error34.1
Cost64
\[0\]
Alternative 13
Error61.9
Cost64
\[1\]

Error

Derivation

  1. Split input into 2 regimes

  2. if l < -1.92884772123948671e98 or 5.6555399113331643e206 < l

    1. Initial program 61.2

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

    2. Simplified59.6

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

    3. Taylor expanded around 0 55.5

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

    4. Simplified55.5

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

    6. Applied associate-*l*_binary64_36053.8

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

    8. Applied times-frac_binary64_42551.0

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

    10. Applied *-un-lft-identity_binary64_41951.0

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

    11. Applied times-frac_binary64_42551.0

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

    12. Applied associate-*r*_binary64_35950.5

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

    13. Simplified9.0

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

    14. Simplified9.0

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

    if -1.92884772123948671e98 < l < 5.6555399113331643e206

    1. Initial program 45.9

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

    2. Simplified36.8

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

    3. Taylor expanded around 0 16.1

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

    4. Simplified16.1

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

    6. Applied associate-*l*_binary64_36013.9

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

    8. Applied times-frac_binary64_42511.9

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

    10. Applied *-un-lft-identity_binary64_41911.9

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

    11. Applied times-frac_binary64_4259.5

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

    12. Applied associate-*l*_binary64_3607.3

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

    13. Simplified7.3

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

    14. Simplified7.3

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

  4. Final simplification7.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.9288477212394867 \cdot 10^{+98} \lor \neg \left(\ell \leq 5.655539911333164 \cdot 10^{+206}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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))))