Average Error: 0.5 → 0.6
Time: 8.5s
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 r296358 = 1.0;
        double r296359 = 5.0;
        double r296360 = v;
        double r296361 = r296360 * r296360;
        double r296362 = r296359 * r296361;
        double r296363 = r296358 - r296362;
        double r296364 = atan2(1.0, 0.0);
        double r296365 = t;
        double r296366 = r296364 * r296365;
        double r296367 = 2.0;
        double r296368 = 3.0;
        double r296369 = r296368 * r296361;
        double r296370 = r296358 - r296369;
        double r296371 = r296367 * r296370;
        double r296372 = sqrt(r296371);
        double r296373 = r296366 * r296372;
        double r296374 = r296358 - r296361;
        double r296375 = r296373 * r296374;
        double r296376 = r296363 / r296375;
        return r296376;
}
↓
double f(double v, double t) {
        double r296377 = 1.5;
        double r296378 = v;
        double r296379 = 2.0;
        double r296380 = pow(r296378, r296379);
        double r296381 = t;
        double r296382 = 2.0;
        double r296383 = sqrt(r296382);
        double r296384 = 1.0;
        double r296385 = sqrt(r296384);
        double r296386 = atan2(1.0, 0.0);
        double r296387 = r296385 * r296386;
        double r296388 = r296383 * r296387;
        double r296389 = r296381 * r296388;
        double r296390 = r296380 / r296389;
        double r296391 = r296383 * r296386;
        double r296392 = r296381 * r296391;
        double r296393 = r296385 / r296392;
        double r296394 = r296384 * r296393;
        double r296395 = 4.0;
        double r296396 = pow(r296378, r296395);
        double r296397 = r296396 / r296389;
        double r296398 = 1.125;
        double r296399 = 3.0;
        double r296400 = pow(r296385, r296399);
        double r296401 = r296400 * r296386;
        double r296402 = r296383 * r296401;
        double r296403 = r296381 * r296402;
        double r296404 = r296396 / r296403;
        double r296405 = 4.0;
        double r296406 = r296380 * r296385;
        double r296407 = r296406 / r296392;
        double r296408 = r296396 * r296385;
        double r296409 = r296408 / r296392;
        double r296410 = r296405 * r296409;
        double r296411 = fma(r296405, r296407, r296410);
        double r296412 = fma(r296398, r296404, r296411);
        double r296413 = fma(r296377, r296397, r296412);
        double r296414 = r296394 - r296413;
        double r296415 = fma(r296377, r296390, r296414);
        return r296415;
}

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 2019362 +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)))))