Average Error: 32.2 → 9.3
Time: 34.7s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7.451781871449174 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(\frac{\frac{t \cdot \sin k}{\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}}}{\sqrt[3]{\frac{\ell}{t}}} \cdot \tan k\right)}{\frac{\ell}{t}}}\\ \mathbf{elif}\;t \le 2.030695630825502 \cdot 10^{-34}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, \frac{t \cdot t}{\frac{\ell \cdot \cos k}{\sin k \cdot \sin k}}, \frac{\sin k \cdot \sin k}{\frac{\ell \cdot \cos k}{k \cdot k}}\right)}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(\frac{\frac{t \cdot \sin k}{\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}}}{\sqrt[3]{\frac{\ell}{t}}} \cdot \tan k\right)}{\frac{\ell}{t}}}\\ \end{array}\]
\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 \le -7.451781871449174 \cdot 10^{-25}:\\
\;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(\frac{\frac{t \cdot \sin k}{\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}}}{\sqrt[3]{\frac{\ell}{t}}} \cdot \tan k\right)}{\frac{\ell}{t}}}\\

\mathbf{elif}\;t \le 2.030695630825502 \cdot 10^{-34}:\\
\;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, \frac{t \cdot t}{\frac{\ell \cdot \cos k}{\sin k \cdot \sin k}}, \frac{\sin k \cdot \sin k}{\frac{\ell \cdot \cos k}{k \cdot k}}\right)}{\frac{\ell}{t}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(\frac{\frac{t \cdot \sin k}{\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}}}{\sqrt[3]{\frac{\ell}{t}}} \cdot \tan k\right)}{\frac{\ell}{t}}}\\

\end{array}
double f(double t, double l, double k) {
        double r1415288 = 2.0;
        double r1415289 = t;
        double r1415290 = 3.0;
        double r1415291 = pow(r1415289, r1415290);
        double r1415292 = l;
        double r1415293 = r1415292 * r1415292;
        double r1415294 = r1415291 / r1415293;
        double r1415295 = k;
        double r1415296 = sin(r1415295);
        double r1415297 = r1415294 * r1415296;
        double r1415298 = tan(r1415295);
        double r1415299 = r1415297 * r1415298;
        double r1415300 = 1.0;
        double r1415301 = r1415295 / r1415289;
        double r1415302 = pow(r1415301, r1415288);
        double r1415303 = r1415300 + r1415302;
        double r1415304 = r1415303 + r1415300;
        double r1415305 = r1415299 * r1415304;
        double r1415306 = r1415288 / r1415305;
        return r1415306;
}
↓
double f(double t, double l, double k) {
        double r1415307 = t;
        double r1415308 = -7.451781871449174e-25;
        bool r1415309 = r1415307 <= r1415308;
        double r1415310 = 2.0;
        double r1415311 = k;
        double r1415312 = r1415311 / r1415307;
        double r1415313 = fma(r1415312, r1415312, r1415310);
        double r1415314 = sin(r1415311);
        double r1415315 = r1415307 * r1415314;
        double r1415316 = l;
        double r1415317 = r1415316 / r1415307;
        double r1415318 = cbrt(r1415317);
        double r1415319 = r1415318 * r1415318;
        double r1415320 = r1415315 / r1415319;
        double r1415321 = r1415320 / r1415318;
        double r1415322 = tan(r1415311);
        double r1415323 = r1415321 * r1415322;
        double r1415324 = r1415313 * r1415323;
        double r1415325 = r1415324 / r1415317;
        double r1415326 = r1415310 / r1415325;
        double r1415327 = 2.030695630825502e-34;
        bool r1415328 = r1415307 <= r1415327;
        double r1415329 = r1415307 * r1415307;
        double r1415330 = cos(r1415311);
        double r1415331 = r1415316 * r1415330;
        double r1415332 = r1415314 * r1415314;
        double r1415333 = r1415331 / r1415332;
        double r1415334 = r1415329 / r1415333;
        double r1415335 = r1415311 * r1415311;
        double r1415336 = r1415331 / r1415335;
        double r1415337 = r1415332 / r1415336;
        double r1415338 = fma(r1415310, r1415334, r1415337);
        double r1415339 = r1415338 / r1415317;
        double r1415340 = r1415310 / r1415339;
        double r1415341 = r1415328 ? r1415340 : r1415326;
        double r1415342 = r1415309 ? r1415326 : r1415341;
        return r1415342;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if t < -7.451781871449174e-25 or 2.030695630825502e-34 < t

    1. Initial program 22.0

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

    2. Simplified12.1

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

    4. Applied *-un-lft-identity12.1

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

    5. Applied times-frac11.4

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

    6. Applied associate-*l*8.4

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{1}{\frac{\ell}{t}} \cdot \left(\frac{t}{\frac{\ell}{t}} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}\]
    7. Using strategy rm

    8. Applied associate-*l/7.0

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

    9. Applied associate-*r/7.0

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

    10. Applied associate-*l/4.1

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

    11. Applied associate-*l/3.7

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

    12. Simplified3.5

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

    14. Applied add-cube-cbrt3.8

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

    15. Applied associate-/r*3.8

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

    if -7.451781871449174e-25 < t < 2.030695630825502e-34

    1. Initial program 52.1

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

    2. Simplified39.1

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

    4. Applied *-un-lft-identity39.1

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

    5. Applied times-frac38.3

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

    6. Applied associate-*l*37.2

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{1}{\frac{\ell}{t}} \cdot \left(\frac{t}{\frac{\ell}{t}} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}\]
    7. Using strategy rm

    8. Applied associate-*l/37.8

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

    9. Applied associate-*r/37.8

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

    10. Applied associate-*l/38.7

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

    11. Applied associate-*l/35.0

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

    12. Simplified35.0

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

    13. Taylor expanded around inf 22.8

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

    14. Simplified20.0

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -7.451781871449174 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(\frac{\frac{t \cdot \sin k}{\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}}}{\sqrt[3]{\frac{\ell}{t}}} \cdot \tan k\right)}{\frac{\ell}{t}}}\\ \mathbf{elif}\;t \le 2.030695630825502 \cdot 10^{-34}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, \frac{t \cdot t}{\frac{\ell \cdot \cos k}{\sin k \cdot \sin k}}, \frac{\sin k \cdot \sin k}{\frac{\ell \cdot \cos k}{k \cdot k}}\right)}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right) \cdot \left(\frac{\frac{t \cdot \sin k}{\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}}}{\sqrt[3]{\frac{\ell}{t}}} \cdot \tan k\right)}{\frac{\ell}{t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))