Falkner and Boettcher, Equation (20:1,3)

Average Error: 0.4 → 0.3

Time: 24.2s

Precision: 64

Math

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

↓

\[\left(\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\left(t \cdot \sqrt{2 \cdot \left({1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}\right)}\right) \cdot \left({\left({1}^{3}\right)}^{3} - {\left({v}^{6}\right)}^{3}\right)} \cdot \left(\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)} \cdot \left({1}^{3} \cdot {1}^{3} + \left({v}^{6} \cdot {v}^{6} + {1}^{3} \cdot {v}^{6}\right)\right)\right)\right) \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\]

C

double f(double v, double t) {
        double r197429 = 1.0;
        double r197430 = 5.0;
        double r197431 = v;
        double r197432 = r197431 * r197431;
        double r197433 = r197430 * r197432;
        double r197434 = r197429 - r197433;
        double r197435 = atan2(1.0, 0.0);
        double r197436 = t;
        double r197437 = r197435 * r197436;
        double r197438 = 2.0;
        double r197439 = 3.0;
        double r197440 = r197439 * r197432;
        double r197441 = r197429 - r197440;
        double r197442 = r197438 * r197441;
        double r197443 = sqrt(r197442);
        double r197444 = r197437 * r197443;
        double r197445 = r197429 - r197432;
        double r197446 = r197444 * r197445;
        double r197447 = r197434 / r197446;
        return r197447;
}

↓

double f(double v, double t) {
        double r197448 = 1.0;
        double r197449 = 5.0;
        double r197450 = v;
        double r197451 = r197450 * r197450;
        double r197452 = r197449 * r197451;
        double r197453 = r197448 - r197452;
        double r197454 = atan2(1.0, 0.0);
        double r197455 = r197453 / r197454;
        double r197456 = t;
        double r197457 = 2.0;
        double r197458 = 3.0;
        double r197459 = pow(r197448, r197458);
        double r197460 = 3.0;
        double r197461 = r197460 * r197451;
        double r197462 = pow(r197461, r197458);
        double r197463 = r197459 - r197462;
        double r197464 = r197457 * r197463;
        double r197465 = sqrt(r197464);
        double r197466 = r197456 * r197465;
        double r197467 = pow(r197459, r197458);
        double r197468 = 6.0;
        double r197469 = pow(r197450, r197468);
        double r197470 = pow(r197469, r197458);
        double r197471 = r197467 - r197470;
        double r197472 = r197466 * r197471;
        double r197473 = r197455 / r197472;
        double r197474 = r197448 * r197448;
        double r197475 = r197461 * r197461;
        double r197476 = r197448 * r197461;
        double r197477 = r197475 + r197476;
        double r197478 = r197474 + r197477;
        double r197479 = sqrt(r197478);
        double r197480 = r197459 * r197459;
        double r197481 = r197469 * r197469;
        double r197482 = r197459 * r197469;
        double r197483 = r197481 + r197482;
        double r197484 = r197480 + r197483;
        double r197485 = r197479 * r197484;
        double r197486 = r197473 * r197485;
        double r197487 = r197451 * r197451;
        double r197488 = r197448 * r197451;
        double r197489 = r197487 + r197488;
        double r197490 = r197474 + r197489;
        double r197491 = r197486 * r197490;
        return r197491;
}

Error

Bits error versus v

Bits error versus t

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 associate-*l* 0.4

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

5. Applied flip3-- 0.4

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

6. Applied associate-*r/ 0.4

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

7. Applied associate-/r/ 0.4

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

8. Simplified 0.3

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

10. Applied flip3-- 0.3

    \[\leadsto \frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \color{blue}{\frac{{\left({1}^{3}\right)}^{3} - {\left({v}^{6}\right)}^{3}}{{1}^{3} \cdot {1}^{3} + \left({v}^{6} \cdot {v}^{6} + {1}^{3} \cdot {v}^{6}\right)}}} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\]

11. Applied flip3-- 0.3

    \[\leadsto \frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\left(t \cdot \sqrt{2 \cdot \color{blue}{\frac{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}}\right) \cdot \frac{{\left({1}^{3}\right)}^{3} - {\left({v}^{6}\right)}^{3}}{{1}^{3} \cdot {1}^{3} + \left({v}^{6} \cdot {v}^{6} + {1}^{3} \cdot {v}^{6}\right)}} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\]

12. Applied associate-*r/ 0.3

    \[\leadsto \frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\left(t \cdot \sqrt{\color{blue}{\frac{2 \cdot \left({1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}\right)}{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}}\right) \cdot \frac{{\left({1}^{3}\right)}^{3} - {\left({v}^{6}\right)}^{3}}{{1}^{3} \cdot {1}^{3} + \left({v}^{6} \cdot {v}^{6} + {1}^{3} \cdot {v}^{6}\right)}} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\]

13. Applied sqrt-div 0.3

    \[\leadsto \frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\left(t \cdot \color{blue}{\frac{\sqrt{2 \cdot \left({1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}\right)}}{\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}}\right) \cdot \frac{{\left({1}^{3}\right)}^{3} - {\left({v}^{6}\right)}^{3}}{{1}^{3} \cdot {1}^{3} + \left({v}^{6} \cdot {v}^{6} + {1}^{3} \cdot {v}^{6}\right)}} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\]

14. Applied associate-*r/ 0.3

    \[\leadsto \frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\color{blue}{\frac{t \cdot \sqrt{2 \cdot \left({1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}\right)}}{\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}} \cdot \frac{{\left({1}^{3}\right)}^{3} - {\left({v}^{6}\right)}^{3}}{{1}^{3} \cdot {1}^{3} + \left({v}^{6} \cdot {v}^{6} + {1}^{3} \cdot {v}^{6}\right)}} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\]

15. Applied frac-times 0.3

    \[\leadsto \frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\color{blue}{\frac{\left(t \cdot \sqrt{2 \cdot \left({1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}\right)}\right) \cdot \left({\left({1}^{3}\right)}^{3} - {\left({v}^{6}\right)}^{3}\right)}{\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)} \cdot \left({1}^{3} \cdot {1}^{3} + \left({v}^{6} \cdot {v}^{6} + {1}^{3} \cdot {v}^{6}\right)\right)}}} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\]

16. Applied associate-/r/ 0.3

    \[\leadsto \color{blue}{\left(\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\left(t \cdot \sqrt{2 \cdot \left({1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}\right)}\right) \cdot \left({\left({1}^{3}\right)}^{3} - {\left({v}^{6}\right)}^{3}\right)} \cdot \left(\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)} \cdot \left({1}^{3} \cdot {1}^{3} + \left({v}^{6} \cdot {v}^{6} + {1}^{3} \cdot {v}^{6}\right)\right)\right)\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\]

17. Final simplification 0.3

    \[\leadsto \left(\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\left(t \cdot \sqrt{2 \cdot \left({1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}\right)}\right) \cdot \left({\left({1}^{3}\right)}^{3} - {\left({v}^{6}\right)}^{3}\right)} \cdot \left(\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)} \cdot \left({1}^{3} \cdot {1}^{3} + \left({v}^{6} \cdot {v}^{6} + {1}^{3} \cdot {v}^{6}\right)\right)\right)\right) \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\]

Reproduce

herbie shell --seed 2019303 
(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)))))