VandenBroeck and Keller, Equation (6)

Average Error: 16.0 → 8.7

Time: 34.8s

Precision: 64

• could not determine a ground truth for program body

  1. F = -2.4802802254510387e-30
  2. l = -4062721766964008.5

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -2.0471110225000456 \cdot 10^{+18}:\\ \;\;\;\;\pi \cdot \ell - \frac{\left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\right)}{F}\\ \mathbf{elif}\;\pi \cdot \ell \le 2.2961577739006424 \cdot 10^{-14}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{1}{F} \cdot \tan \left(\pi \cdot \ell\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\right)}{F}\\ \end{array}\]

C

double f(double F, double l) {
        double r560352 = atan2(1.0, 0.0);
        double r560353 = l;
        double r560354 = r560352 * r560353;
        double r560355 = 1.0;
        double r560356 = F;
        double r560357 = r560356 * r560356;
        double r560358 = r560355 / r560357;
        double r560359 = tan(r560354);
        double r560360 = r560358 * r560359;
        double r560361 = r560354 - r560360;
        return r560361;
}

↓

double f(double F, double l) {
        double r560362 = atan2(1.0, 0.0);
        double r560363 = l;
        double r560364 = r560362 * r560363;
        double r560365 = -2.0471110225000456e+18;
        bool r560366 = r560364 <= r560365;
        double r560367 = tan(r560364);
        double r560368 = F;
        double r560369 = r560367 / r560368;
        double r560370 = /* ERROR: no posit support in C */;
        double r560371 = /* ERROR: no posit support in C */;
        double r560372 = r560371 / r560368;
        double r560373 = r560364 - r560372;
        double r560374 = 2.2961577739006424e-14;
        bool r560375 = r560364 <= r560374;
        double r560376 = 1.0;
        double r560377 = r560376 / r560368;
        double r560378 = r560377 * r560367;
        double r560379 = r560378 / r560368;
        double r560380 = r560364 - r560379;
        double r560381 = r560375 ? r560380 : r560373;
        double r560382 = r560366 ? r560373 : r560381;
        return r560382;
}

Error

Bits error versus F

Bits error versus l

Derivation

1. Split input into 2 regimes

2. if (* PI l) < -2.0471110225000456e+18 or 2.2961577739006424e-14 < (* PI l)

   1. Initial program 22.7

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

   2. Simplified 22.7

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}}\]
   3. Using strategy rm

   4. Applied insert-posit16 16.0

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

   if -2.0471110225000456e+18 < (* PI l) < 2.2961577739006424e-14

   1. Initial program 8.6

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

   2. Simplified 0.6

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}}\]
   3. Using strategy rm

   4. Applied div-inv 0.7

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F}}}{F}\]
3. Recombined 2 regimes into one program.

4. Final simplification 8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -2.0471110225000456 \cdot 10^{+18}:\\ \;\;\;\;\pi \cdot \ell - \frac{\left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\right)}{F}\\ \mathbf{elif}\;\pi \cdot \ell \le 2.2961577739006424 \cdot 10^{-14}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{1}{F} \cdot \tan \left(\pi \cdot \ell\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\left(\left(\frac{\tan \left(\pi \cdot \ell\right)}{F}\right)\right)}{F}\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))