Average Error: 0.4 → 0.3
Time: 8.5s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{t}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)}\]
\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)}
↓
\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{t}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)}
double f(double v, double t) {
        double r279471 = 1.0;
        double r279472 = 5.0;
        double r279473 = v;
        double r279474 = r279473 * r279473;
        double r279475 = r279472 * r279474;
        double r279476 = r279471 - r279475;
        double r279477 = atan2(1.0, 0.0);
        double r279478 = t;
        double r279479 = r279477 * r279478;
        double r279480 = 2.0;
        double r279481 = 3.0;
        double r279482 = r279481 * r279474;
        double r279483 = r279471 - r279482;
        double r279484 = r279480 * r279483;
        double r279485 = sqrt(r279484);
        double r279486 = r279479 * r279485;
        double r279487 = r279471 - r279474;
        double r279488 = r279486 * r279487;
        double r279489 = r279476 / r279488;
        return r279489;
}
↓
double f(double v, double t) {
        double r279490 = 1.0;
        double r279491 = 5.0;
        double r279492 = v;
        double r279493 = r279492 * r279492;
        double r279494 = r279491 * r279493;
        double r279495 = r279490 - r279494;
        double r279496 = atan2(1.0, 0.0);
        double r279497 = r279495 / r279496;
        double r279498 = t;
        double r279499 = r279497 / r279498;
        double r279500 = 2.0;
        double r279501 = 3.0;
        double r279502 = r279501 * r279493;
        double r279503 = r279490 - r279502;
        double r279504 = r279500 * r279503;
        double r279505 = sqrt(r279504);
        double r279506 = r279490 - r279493;
        double r279507 = r279505 * r279506;
        double r279508 = r279499 / r279507;
        return r279508;
}

Error

Bits error versus v

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.4

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

  3. Applied flip3--0.4

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

  4. Applied associate-/l/0.4

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

  5. Simplified0.4

    \[\leadsto \frac{{1}^{3} - {\left(5 \cdot \left(v \cdot v\right)\right)}^{3}}{\color{blue}{\left(\mathsf{fma}\left(5 \cdot \left(v \cdot v\right), 1 + 5 \cdot \left(v \cdot v\right), 1 \cdot 1\right) \cdot \left(\pi \cdot t\right)\right) \cdot \left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)\right)}}\]
  6. Using strategy rm

  7. Applied associate-/r*0.4

    \[\leadsto \color{blue}{\frac{\frac{{1}^{3} - {\left(5 \cdot \left(v \cdot v\right)\right)}^{3}}{\mathsf{fma}\left(5 \cdot \left(v \cdot v\right), 1 + 5 \cdot \left(v \cdot v\right), 1 \cdot 1\right) \cdot \left(\pi \cdot t\right)}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)}}\]

  8. Taylor expanded around 0 0.4

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

  9. Simplified0.4

    \[\leadsto \frac{\color{blue}{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot t}}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)}\]
  10. Using strategy rm

  11. Applied associate-/r*0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2020059 +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)))))