Average Error: 0.4 → 0.1
Time: 23.5s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\left(1 - \left(5 \cdot v\right) \cdot v\right) \cdot \frac{\frac{1}{\pi \cdot \sqrt{\left(1 \cdot 1 - \left(\left(v \cdot v\right) \cdot 3\right) \cdot \left(\left(v \cdot v\right) \cdot 3\right)\right) \cdot 2}}}{t}}{\left(1 \cdot 1\right) \cdot 1 - \left(\left(v \cdot v\right) \cdot v\right) \cdot \left(\left(v \cdot v\right) \cdot v\right)} \cdot \left(\left(\left(1 \cdot \left(v \cdot v\right) + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) + 1 \cdot 1\right) \cdot \sqrt{\log \left(e^{\left(v \cdot v\right) \cdot 3}\right) + 1}\right)\]
\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{\left(1 - \left(5 \cdot v\right) \cdot v\right) \cdot \frac{\frac{1}{\pi \cdot \sqrt{\left(1 \cdot 1 - \left(\left(v \cdot v\right) \cdot 3\right) \cdot \left(\left(v \cdot v\right) \cdot 3\right)\right) \cdot 2}}}{t}}{\left(1 \cdot 1\right) \cdot 1 - \left(\left(v \cdot v\right) \cdot v\right) \cdot \left(\left(v \cdot v\right) \cdot v\right)} \cdot \left(\left(\left(1 \cdot \left(v \cdot v\right) + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) + 1 \cdot 1\right) \cdot \sqrt{\log \left(e^{\left(v \cdot v\right) \cdot 3}\right) + 1}\right)
double f(double v, double t) {
        double r7406492 = 1.0;
        double r7406493 = 5.0;
        double r7406494 = v;
        double r7406495 = r7406494 * r7406494;
        double r7406496 = r7406493 * r7406495;
        double r7406497 = r7406492 - r7406496;
        double r7406498 = atan2(1.0, 0.0);
        double r7406499 = t;
        double r7406500 = r7406498 * r7406499;
        double r7406501 = 2.0;
        double r7406502 = 3.0;
        double r7406503 = r7406502 * r7406495;
        double r7406504 = r7406492 - r7406503;
        double r7406505 = r7406501 * r7406504;
        double r7406506 = sqrt(r7406505);
        double r7406507 = r7406500 * r7406506;
        double r7406508 = r7406492 - r7406495;
        double r7406509 = r7406507 * r7406508;
        double r7406510 = r7406497 / r7406509;
        return r7406510;
}
↓
double f(double v, double t) {
        double r7406511 = 1.0;
        double r7406512 = 5.0;
        double r7406513 = v;
        double r7406514 = r7406512 * r7406513;
        double r7406515 = r7406514 * r7406513;
        double r7406516 = r7406511 - r7406515;
        double r7406517 = 1.0;
        double r7406518 = atan2(1.0, 0.0);
        double r7406519 = r7406511 * r7406511;
        double r7406520 = r7406513 * r7406513;
        double r7406521 = 3.0;
        double r7406522 = r7406520 * r7406521;
        double r7406523 = r7406522 * r7406522;
        double r7406524 = r7406519 - r7406523;
        double r7406525 = 2.0;
        double r7406526 = r7406524 * r7406525;
        double r7406527 = sqrt(r7406526);
        double r7406528 = r7406518 * r7406527;
        double r7406529 = r7406517 / r7406528;
        double r7406530 = t;
        double r7406531 = r7406529 / r7406530;
        double r7406532 = r7406516 * r7406531;
        double r7406533 = r7406519 * r7406511;
        double r7406534 = r7406520 * r7406513;
        double r7406535 = r7406534 * r7406534;
        double r7406536 = r7406533 - r7406535;
        double r7406537 = r7406532 / r7406536;
        double r7406538 = r7406511 * r7406520;
        double r7406539 = r7406520 * r7406520;
        double r7406540 = r7406538 + r7406539;
        double r7406541 = r7406540 + r7406519;
        double r7406542 = exp(r7406522);
        double r7406543 = log(r7406542);
        double r7406544 = r7406543 + r7406511;
        double r7406545 = sqrt(r7406544);
        double r7406546 = r7406541 * r7406545;
        double r7406547 = r7406537 * r7406546;
        return r7406547;
}

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

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

  4. Applied flip--0.4

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

  5. Applied associate-*r/0.4

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

  6. Applied sqrt-div0.5

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

  7. Applied associate-*r/0.5

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

  8. Applied frac-times0.4

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

  9. Applied associate-/r/0.5

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

  10. Simplified0.4

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

  12. Applied div-inv0.4

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

  13. Simplified0.1

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

  15. Applied add-log-exp0.1

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

  16. Final simplification0.1

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1.0 (* 5.0 (* v v))) (* (* (* PI t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))