Average Error: 0.5 → 0.6
Time: 8.9s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\mathsf{fma}\left(1.5, \frac{{v}^{2}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)}, 1 \cdot \frac{\sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} - \mathsf{fma}\left(1.5, \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)}, \mathsf{fma}\left(1.125, \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left({\left(\sqrt{1}\right)}^{3} \cdot \pi\right)\right)}, \mathsf{fma}\left(4, \frac{{v}^{2} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}, 4 \cdot \frac{{v}^{4} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)\right)\right)\right)\]
\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}
↓
\mathsf{fma}\left(1.5, \frac{{v}^{2}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)}, 1 \cdot \frac{\sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} - \mathsf{fma}\left(1.5, \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)}, \mathsf{fma}\left(1.125, \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left({\left(\sqrt{1}\right)}^{3} \cdot \pi\right)\right)}, \mathsf{fma}\left(4, \frac{{v}^{2} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}, 4 \cdot \frac{{v}^{4} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)\right)\right)\right)
double f(double v, double t) {
        double r260341 = 1.0;
        double r260342 = 5.0;
        double r260343 = v;
        double r260344 = r260343 * r260343;
        double r260345 = r260342 * r260344;
        double r260346 = r260341 - r260345;
        double r260347 = atan2(1.0, 0.0);
        double r260348 = t;
        double r260349 = r260347 * r260348;
        double r260350 = 2.0;
        double r260351 = 3.0;
        double r260352 = r260351 * r260344;
        double r260353 = r260341 - r260352;
        double r260354 = r260350 * r260353;
        double r260355 = sqrt(r260354);
        double r260356 = r260349 * r260355;
        double r260357 = r260341 - r260344;
        double r260358 = r260356 * r260357;
        double r260359 = r260346 / r260358;
        return r260359;
}
↓
double f(double v, double t) {
        double r260360 = 1.5;
        double r260361 = v;
        double r260362 = 2.0;
        double r260363 = pow(r260361, r260362);
        double r260364 = t;
        double r260365 = 2.0;
        double r260366 = sqrt(r260365);
        double r260367 = 1.0;
        double r260368 = sqrt(r260367);
        double r260369 = atan2(1.0, 0.0);
        double r260370 = r260368 * r260369;
        double r260371 = r260366 * r260370;
        double r260372 = r260364 * r260371;
        double r260373 = r260363 / r260372;
        double r260374 = r260366 * r260369;
        double r260375 = r260364 * r260374;
        double r260376 = r260368 / r260375;
        double r260377 = r260367 * r260376;
        double r260378 = 4.0;
        double r260379 = pow(r260361, r260378);
        double r260380 = r260379 / r260372;
        double r260381 = 1.125;
        double r260382 = 3.0;
        double r260383 = pow(r260368, r260382);
        double r260384 = r260383 * r260369;
        double r260385 = r260366 * r260384;
        double r260386 = r260364 * r260385;
        double r260387 = r260379 / r260386;
        double r260388 = 4.0;
        double r260389 = r260363 * r260368;
        double r260390 = r260389 / r260375;
        double r260391 = r260379 * r260368;
        double r260392 = r260391 / r260375;
        double r260393 = r260388 * r260392;
        double r260394 = fma(r260388, r260390, r260393);
        double r260395 = fma(r260381, r260387, r260394);
        double r260396 = fma(r260360, r260380, r260395);
        double r260397 = r260377 - r260396;
        double r260398 = fma(r260360, r260373, r260397);
        return r260398;
}

Error

Bits error versus v

Bits error versus t

Derivation

  1. Initial program 0.5

    \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]

  2. Taylor expanded around 0 0.6

    \[\leadsto \color{blue}{\left(1.5 \cdot \frac{{v}^{2}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)} + 1 \cdot \frac{\sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right) - \left(1.5 \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)} + \left(1.125 \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left({\left(\sqrt{1}\right)}^{3} \cdot \pi\right)\right)} + \left(4 \cdot \frac{{v}^{2} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + 4 \cdot \frac{{v}^{4} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)\right)\right)}\]

  3. Simplified0.6

    \[\leadsto \color{blue}{\mathsf{fma}\left(1.5, \frac{{v}^{2}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)}, 1 \cdot \frac{\sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} - \mathsf{fma}\left(1.5, \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)}, \mathsf{fma}\left(1.125, \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left({\left(\sqrt{1}\right)}^{3} \cdot \pi\right)\right)}, \mathsf{fma}\left(4, \frac{{v}^{2} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}, 4 \cdot \frac{{v}^{4} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)\right)\right)\right)}\]

  4. Final simplification0.6

    \[\leadsto \mathsf{fma}\left(1.5, \frac{{v}^{2}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)}, 1 \cdot \frac{\sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} - \mathsf{fma}\left(1.5, \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)}, \mathsf{fma}\left(1.125, \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left({\left(\sqrt{1}\right)}^{3} \cdot \pi\right)\right)}, \mathsf{fma}\left(4, \frac{{v}^{2} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}, 4 \cdot \frac{{v}^{4} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2019344 +o rules:numerics
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  :precision binary64
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))