Average Error: 0.5 → 0.4
Time: 23.6s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{1 \cdot 1 - \left(5 \cdot 5\right) \cdot {v}^{4}}{\sqrt{\left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right) \cdot 2}}}{t}}{\pi} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, 3, 1\right)}}{\left(1 - v \cdot v\right) \cdot \mathsf{fma}\left(5, v \cdot v, 1\right)}\]
\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 \cdot 1 - \left(5 \cdot 5\right) \cdot {v}^{4}}{\sqrt{\left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right) \cdot 2}}}{t}}{\pi} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, 3, 1\right)}}{\left(1 - v \cdot v\right) \cdot \mathsf{fma}\left(5, v \cdot v, 1\right)}
double f(double v, double t) {
        double r181514 = 1.0;
        double r181515 = 5.0;
        double r181516 = v;
        double r181517 = r181516 * r181516;
        double r181518 = r181515 * r181517;
        double r181519 = r181514 - r181518;
        double r181520 = atan2(1.0, 0.0);
        double r181521 = t;
        double r181522 = r181520 * r181521;
        double r181523 = 2.0;
        double r181524 = 3.0;
        double r181525 = r181524 * r181517;
        double r181526 = r181514 - r181525;
        double r181527 = r181523 * r181526;
        double r181528 = sqrt(r181527);
        double r181529 = r181522 * r181528;
        double r181530 = r181514 - r181517;
        double r181531 = r181529 * r181530;
        double r181532 = r181519 / r181531;
        return r181532;
}
↓
double f(double v, double t) {
        double r181533 = 1.0;
        double r181534 = r181533 * r181533;
        double r181535 = 5.0;
        double r181536 = r181535 * r181535;
        double r181537 = v;
        double r181538 = 4.0;
        double r181539 = pow(r181537, r181538);
        double r181540 = r181536 * r181539;
        double r181541 = r181534 - r181540;
        double r181542 = 3.0;
        double r181543 = r181542 * r181542;
        double r181544 = r181543 * r181539;
        double r181545 = r181534 - r181544;
        double r181546 = 2.0;
        double r181547 = r181545 * r181546;
        double r181548 = sqrt(r181547);
        double r181549 = r181541 / r181548;
        double r181550 = t;
        double r181551 = r181549 / r181550;
        double r181552 = atan2(1.0, 0.0);
        double r181553 = r181551 / r181552;
        double r181554 = r181537 * r181537;
        double r181555 = fma(r181554, r181542, r181533);
        double r181556 = sqrt(r181555);
        double r181557 = r181533 - r181554;
        double r181558 = fma(r181535, r181554, r181533);
        double r181559 = r181557 * r181558;
        double r181560 = r181556 / r181559;
        double r181561 = r181553 * r181560;
        return r181561;
}

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. Using strategy rm

  3. Applied flip--0.5

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

  4. Applied associate-/l/0.5

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

  5. Simplified0.5

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

  7. Applied associate-/r*0.4

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

  8. Simplified0.4

    \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \left(5 \cdot 5\right) \cdot {v}^{4}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}}}{\left(t \cdot \pi\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \mathsf{fma}\left(5, v \cdot v, 1\right)\right)}\]
  9. Using strategy rm

  10. Applied flip--0.4

    \[\leadsto \frac{\frac{1 \cdot 1 - \left(5 \cdot 5\right) \cdot {v}^{4}}{\sqrt{2 \cdot \color{blue}{\frac{1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)}{1 + 3 \cdot \left(v \cdot v\right)}}}}}{\left(t \cdot \pi\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \mathsf{fma}\left(5, v \cdot v, 1\right)\right)}\]

  11. Applied associate-*r/0.4

    \[\leadsto \frac{\frac{1 \cdot 1 - \left(5 \cdot 5\right) \cdot {v}^{4}}{\sqrt{\color{blue}{\frac{2 \cdot \left(1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}{1 + 3 \cdot \left(v \cdot v\right)}}}}}{\left(t \cdot \pi\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \mathsf{fma}\left(5, v \cdot v, 1\right)\right)}\]

  12. Applied sqrt-div0.4

    \[\leadsto \frac{\frac{1 \cdot 1 - \left(5 \cdot 5\right) \cdot {v}^{4}}{\color{blue}{\frac{\sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}{\sqrt{1 + 3 \cdot \left(v \cdot v\right)}}}}}{\left(t \cdot \pi\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \mathsf{fma}\left(5, v \cdot v, 1\right)\right)}\]

  13. Applied associate-/r/0.4

    \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \left(5 \cdot 5\right) \cdot {v}^{4}}{\sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}}}{\left(t \cdot \pi\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \mathsf{fma}\left(5, v \cdot v, 1\right)\right)}\]

  14. Applied times-frac0.4

    \[\leadsto \color{blue}{\frac{\frac{1 \cdot 1 - \left(5 \cdot 5\right) \cdot {v}^{4}}{\sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}}{t \cdot \pi} \cdot \frac{\sqrt{1 + 3 \cdot \left(v \cdot v\right)}}{\left(1 - v \cdot v\right) \cdot \mathsf{fma}\left(5, v \cdot v, 1\right)}}\]

  15. Simplified0.4

    \[\leadsto \color{blue}{\frac{\frac{\frac{1 \cdot 1 - \left(5 \cdot 5\right) \cdot {v}^{4}}{\sqrt{\left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right) \cdot 2}}}{t}}{\pi}} \cdot \frac{\sqrt{1 + 3 \cdot \left(v \cdot v\right)}}{\left(1 - v \cdot v\right) \cdot \mathsf{fma}\left(5, v \cdot v, 1\right)}\]

  16. Simplified0.4

    \[\leadsto \frac{\frac{\frac{1 \cdot 1 - \left(5 \cdot 5\right) \cdot {v}^{4}}{\sqrt{\left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right) \cdot 2}}}{t}}{\pi} \cdot \color{blue}{\frac{\sqrt{\mathsf{fma}\left(v \cdot v, 3, 1\right)}}{\left(1 - v \cdot v\right) \cdot \mathsf{fma}\left(5, v \cdot v, 1\right)}}\]

  17. Final simplification0.4

    \[\leadsto \frac{\frac{\frac{1 \cdot 1 - \left(5 \cdot 5\right) \cdot {v}^{4}}{\sqrt{\left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right) \cdot 2}}}{t}}{\pi} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, 3, 1\right)}}{\left(1 - v \cdot v\right) \cdot \mathsf{fma}\left(5, v \cdot v, 1\right)}\]

Reproduce

herbie shell --seed 2019212 +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)))))