Average Error: 0.4 → 0.1
Time: 21.8s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\sqrt{\left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right) \cdot 2}}}{t} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}\]
\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{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\sqrt{\left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right) \cdot 2}}}{t} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}
double f(double v, double t) {
        double r222426 = 1.0;
        double r222427 = 5.0;
        double r222428 = v;
        double r222429 = r222428 * r222428;
        double r222430 = r222427 * r222429;
        double r222431 = r222426 - r222430;
        double r222432 = atan2(1.0, 0.0);
        double r222433 = t;
        double r222434 = r222432 * r222433;
        double r222435 = 2.0;
        double r222436 = 3.0;
        double r222437 = r222436 * r222429;
        double r222438 = r222426 - r222437;
        double r222439 = r222435 * r222438;
        double r222440 = sqrt(r222439);
        double r222441 = r222434 * r222440;
        double r222442 = r222426 - r222429;
        double r222443 = r222441 * r222442;
        double r222444 = r222431 / r222443;
        return r222444;
}
↓
double f(double v, double t) {
        double r222445 = 1.0;
        double r222446 = 5.0;
        double r222447 = v;
        double r222448 = r222447 * r222447;
        double r222449 = r222446 * r222448;
        double r222450 = r222445 - r222449;
        double r222451 = atan2(1.0, 0.0);
        double r222452 = r222450 / r222451;
        double r222453 = r222445 * r222445;
        double r222454 = 3.0;
        double r222455 = r222454 * r222454;
        double r222456 = 4.0;
        double r222457 = pow(r222447, r222456);
        double r222458 = r222455 * r222457;
        double r222459 = r222453 - r222458;
        double r222460 = 2.0;
        double r222461 = r222459 * r222460;
        double r222462 = sqrt(r222461);
        double r222463 = r222452 / r222462;
        double r222464 = t;
        double r222465 = r222463 / r222464;
        double r222466 = r222454 * r222448;
        double r222467 = r222445 + r222466;
        double r222468 = sqrt(r222467);
        double r222469 = r222465 * r222468;
        double r222470 = r222445 - r222448;
        double r222471 = r222469 / r222470;
        return r222471;
}

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 associate-/r*0.4

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

  5. Applied associate-*l*0.4

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

  7. Applied flip--0.4

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

  8. Applied associate-*r/0.4

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

  9. Applied sqrt-div0.4

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

  10. Applied associate-*r/0.4

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

  11. Applied associate-*r/0.4

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

  12. Applied associate-/r/0.4

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

  13. Simplified0.3

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

  15. Applied associate-/r*0.1

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

  16. Final simplification0.1

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

Reproduce

herbie shell --seed 2019325 
(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)))))