Average Error: 1.0 → 0.0
Time: 6.2m
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{1}{\frac{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}{\frac{\frac{4}{3}}{\pi - \sqrt{\pi \cdot \left(v \cdot v\right)} \cdot \sqrt{\pi \cdot \left(v \cdot v\right)}}}}\]
\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
↓
\frac{1}{\frac{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}{\frac{\frac{4}{3}}{\pi - \sqrt{\pi \cdot \left(v \cdot v\right)} \cdot \sqrt{\pi \cdot \left(v \cdot v\right)}}}}
double f(double v) {
        double r77475456 = 4.0;
        double r77475457 = 3.0;
        double r77475458 = atan2(1.0, 0.0);
        double r77475459 = r77475457 * r77475458;
        double r77475460 = 1.0;
        double r77475461 = v;
        double r77475462 = r77475461 * r77475461;
        double r77475463 = r77475460 - r77475462;
        double r77475464 = r77475459 * r77475463;
        double r77475465 = 2.0;
        double r77475466 = 6.0;
        double r77475467 = r77475466 * r77475462;
        double r77475468 = r77475465 - r77475467;
        double r77475469 = sqrt(r77475468);
        double r77475470 = r77475464 * r77475469;
        double r77475471 = r77475456 / r77475470;
        return r77475471;
}
↓
double f(double v) {
        double r77475472 = 1.0;
        double r77475473 = 2.0;
        double r77475474 = v;
        double r77475475 = -6.0;
        double r77475476 = r77475474 * r77475475;
        double r77475477 = r77475476 * r77475474;
        double r77475478 = r77475473 + r77475477;
        double r77475479 = sqrt(r77475478);
        double r77475480 = 1.3333333333333333;
        double r77475481 = atan2(1.0, 0.0);
        double r77475482 = r77475474 * r77475474;
        double r77475483 = r77475481 * r77475482;
        double r77475484 = sqrt(r77475483);
        double r77475485 = r77475484 * r77475484;
        double r77475486 = r77475481 - r77475485;
        double r77475487 = r77475480 / r77475486;
        double r77475488 = r77475479 / r77475487;
        double r77475489 = r77475472 / r77475488;
        return r77475489;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.0

    \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{\frac{\frac{4}{3}}{\pi - \left(v \cdot v\right) \cdot \pi}}{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity0.0

    \[\leadsto \frac{\frac{\frac{4}{3}}{\color{blue}{1 \cdot \left(\pi - \left(v \cdot v\right) \cdot \pi\right)}}}{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}\]

  5. Applied *-un-lft-identity0.0

    \[\leadsto \frac{\frac{\color{blue}{1 \cdot \frac{4}{3}}}{1 \cdot \left(\pi - \left(v \cdot v\right) \cdot \pi\right)}}{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}\]

  6. Applied times-frac0.0

    \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{\frac{4}{3}}{\pi - \left(v \cdot v\right) \cdot \pi}}}{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}\]

  7. Applied associate-/l*0.0

    \[\leadsto \color{blue}{\frac{\frac{1}{1}}{\frac{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}{\frac{\frac{4}{3}}{\pi - \left(v \cdot v\right) \cdot \pi}}}}\]

  8. Simplified0.0

    \[\leadsto \frac{\color{blue}{1}}{\frac{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}{\frac{\frac{4}{3}}{\pi - \left(v \cdot v\right) \cdot \pi}}}\]
  9. Using strategy rm

  10. Applied add-sqr-sqrt0.0

    \[\leadsto \frac{1}{\frac{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}{\frac{\frac{4}{3}}{\pi - \color{blue}{\sqrt{\left(v \cdot v\right) \cdot \pi} \cdot \sqrt{\left(v \cdot v\right) \cdot \pi}}}}}\]

  11. Final simplification0.0

    \[\leadsto \frac{1}{\frac{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}{\frac{\frac{4}{3}}{\pi - \sqrt{\pi \cdot \left(v \cdot v\right)} \cdot \sqrt{\pi \cdot \left(v \cdot v\right)}}}}\]

Reproduce

herbie shell --seed 2019107 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))