Average Error: 16.8 → 8.5
Time: 9.8s
Precision: 64
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -9.85292046734543611 \cdot 10^{161}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sin \left(\left(\pi \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)\right) \cdot \sqrt[3]{\ell}\right) \cdot \sqrt{1}}{\cos \left(\pi \cdot \ell\right) \cdot F}\\ \mathbf{elif}\;\pi \cdot \ell \le 3.5124574908168376 \cdot 10^{148}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt{1}}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right) \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \left(\sqrt{\frac{1}{F \cdot F}} \cdot \sqrt{\frac{1}{F \cdot F}}\right) \cdot \tan \left(\pi \cdot \ell\right)\\ \end{array}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\begin{array}{l}
\mathbf{if}\;\pi \cdot \ell \le -9.85292046734543611 \cdot 10^{161}:\\
\;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sin \left(\left(\pi \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)\right) \cdot \sqrt[3]{\ell}\right) \cdot \sqrt{1}}{\cos \left(\pi \cdot \ell\right) \cdot F}\\

\mathbf{elif}\;\pi \cdot \ell \le 3.5124574908168376 \cdot 10^{148}:\\
\;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt{1}}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right) \cdot F}\\

\mathbf{else}:\\
\;\;\;\;\pi \cdot \ell - \left(\sqrt{\frac{1}{F \cdot F}} \cdot \sqrt{\frac{1}{F \cdot F}}\right) \cdot \tan \left(\pi \cdot \ell\right)\\

\end{array}
double f(double F, double l) {
        double r20299 = atan2(1.0, 0.0);
        double r20300 = l;
        double r20301 = r20299 * r20300;
        double r20302 = 1.0;
        double r20303 = F;
        double r20304 = r20303 * r20303;
        double r20305 = r20302 / r20304;
        double r20306 = tan(r20301);
        double r20307 = r20305 * r20306;
        double r20308 = r20301 - r20307;
        return r20308;
}
↓
double f(double F, double l) {
        double r20309 = atan2(1.0, 0.0);
        double r20310 = l;
        double r20311 = r20309 * r20310;
        double r20312 = -9.852920467345436e+161;
        bool r20313 = r20311 <= r20312;
        double r20314 = 1.0;
        double r20315 = sqrt(r20314);
        double r20316 = F;
        double r20317 = r20315 / r20316;
        double r20318 = cbrt(r20310);
        double r20319 = r20318 * r20318;
        double r20320 = r20309 * r20319;
        double r20321 = r20320 * r20318;
        double r20322 = sin(r20321);
        double r20323 = r20322 * r20315;
        double r20324 = cos(r20311);
        double r20325 = r20324 * r20316;
        double r20326 = r20323 / r20325;
        double r20327 = r20317 * r20326;
        double r20328 = r20311 - r20327;
        double r20329 = 3.5124574908168376e+148;
        bool r20330 = r20311 <= r20329;
        double r20331 = sin(r20311);
        double r20332 = r20331 * r20315;
        double r20333 = 0.041666666666666664;
        double r20334 = 4.0;
        double r20335 = pow(r20309, r20334);
        double r20336 = pow(r20310, r20334);
        double r20337 = r20335 * r20336;
        double r20338 = r20333 * r20337;
        double r20339 = 1.0;
        double r20340 = r20338 + r20339;
        double r20341 = 0.5;
        double r20342 = 2.0;
        double r20343 = pow(r20309, r20342);
        double r20344 = pow(r20310, r20342);
        double r20345 = r20343 * r20344;
        double r20346 = r20341 * r20345;
        double r20347 = r20340 - r20346;
        double r20348 = r20347 * r20316;
        double r20349 = r20332 / r20348;
        double r20350 = r20317 * r20349;
        double r20351 = r20311 - r20350;
        double r20352 = r20316 * r20316;
        double r20353 = r20314 / r20352;
        double r20354 = sqrt(r20353);
        double r20355 = r20354 * r20354;
        double r20356 = tan(r20311);
        double r20357 = r20355 * r20356;
        double r20358 = r20311 - r20357;
        double r20359 = r20330 ? r20351 : r20358;
        double r20360 = r20313 ? r20328 : r20359;
        return r20360;
}

Error

Bits error versus F

Bits error versus l

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (* PI l) < -9.852920467345436e+161

    1. Initial program 19.8

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt19.8

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]

    4. Applied times-frac19.8

      \[\leadsto \pi \cdot \ell - \color{blue}{\left(\frac{\sqrt{1}}{F} \cdot \frac{\sqrt{1}}{F}\right)} \cdot \tan \left(\pi \cdot \ell\right)\]

    5. Applied associate-*l*19.8

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\sqrt{1}}{F} \cdot \left(\frac{\sqrt{1}}{F} \cdot \tan \left(\pi \cdot \ell\right)\right)}\]

    6. Taylor expanded around inf 19.8

      \[\leadsto \pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt{1}}{\cos \left(\pi \cdot \ell\right) \cdot F}}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt19.8

      \[\leadsto \pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sin \left(\pi \cdot \color{blue}{\left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}\right)}\right) \cdot \sqrt{1}}{\cos \left(\pi \cdot \ell\right) \cdot F}\]

    9. Applied associate-*r*19.8

      \[\leadsto \pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sin \color{blue}{\left(\left(\pi \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)\right) \cdot \sqrt[3]{\ell}\right)} \cdot \sqrt{1}}{\cos \left(\pi \cdot \ell\right) \cdot F}\]

    if -9.852920467345436e+161 < (* PI l) < 3.5124574908168376e+148

    1. Initial program 15.8

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt15.8

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]

    4. Applied times-frac15.8

      \[\leadsto \pi \cdot \ell - \color{blue}{\left(\frac{\sqrt{1}}{F} \cdot \frac{\sqrt{1}}{F}\right)} \cdot \tan \left(\pi \cdot \ell\right)\]

    5. Applied associate-*l*9.9

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\sqrt{1}}{F} \cdot \left(\frac{\sqrt{1}}{F} \cdot \tan \left(\pi \cdot \ell\right)\right)}\]

    6. Taylor expanded around inf 9.8

      \[\leadsto \pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt{1}}{\cos \left(\pi \cdot \ell\right) \cdot F}}\]

    7. Taylor expanded around 0 4.4

      \[\leadsto \pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt{1}}{\color{blue}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)} \cdot F}\]

    if 3.5124574908168376e+148 < (* PI l)

    1. Initial program 19.4

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt19.4

      \[\leadsto \pi \cdot \ell - \color{blue}{\left(\sqrt{\frac{1}{F \cdot F}} \cdot \sqrt{\frac{1}{F \cdot F}}\right)} \cdot \tan \left(\pi \cdot \ell\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -9.85292046734543611 \cdot 10^{161}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sin \left(\left(\pi \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)\right) \cdot \sqrt[3]{\ell}\right) \cdot \sqrt{1}}{\cos \left(\pi \cdot \ell\right) \cdot F}\\ \mathbf{elif}\;\pi \cdot \ell \le 3.5124574908168376 \cdot 10^{148}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sqrt{1}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right) \cdot \sqrt{1}}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right) \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \left(\sqrt{\frac{1}{F \cdot F}} \cdot \sqrt{\frac{1}{F \cdot F}}\right) \cdot \tan \left(\pi \cdot \ell\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))