Average Error: 1.0 → 0.0
Time: 1.3m
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\sqrt{\frac{4}{3}}}{\pi} \cdot \frac{\sqrt{\frac{4}{3}}}{\sqrt{-6 \cdot \left(v \cdot v\right) + 2} - \sqrt{-6 \cdot \left(v \cdot v\right) + 2} \cdot \left(v \cdot v\right)}\]
\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{\sqrt{\frac{4}{3}}}{\pi} \cdot \frac{\sqrt{\frac{4}{3}}}{\sqrt{-6 \cdot \left(v \cdot v\right) + 2} - \sqrt{-6 \cdot \left(v \cdot v\right) + 2} \cdot \left(v \cdot v\right)}
double f(double v) {
        double r40453467 = 4.0;
        double r40453468 = 3.0;
        double r40453469 = atan2(1.0, 0.0);
        double r40453470 = r40453468 * r40453469;
        double r40453471 = 1.0;
        double r40453472 = v;
        double r40453473 = r40453472 * r40453472;
        double r40453474 = r40453471 - r40453473;
        double r40453475 = r40453470 * r40453474;
        double r40453476 = 2.0;
        double r40453477 = 6.0;
        double r40453478 = r40453477 * r40453473;
        double r40453479 = r40453476 - r40453478;
        double r40453480 = sqrt(r40453479);
        double r40453481 = r40453475 * r40453480;
        double r40453482 = r40453467 / r40453481;
        return r40453482;
}
↓
double f(double v) {
        double r40453483 = 1.3333333333333333;
        double r40453484 = sqrt(r40453483);
        double r40453485 = atan2(1.0, 0.0);
        double r40453486 = r40453484 / r40453485;
        double r40453487 = -6.0;
        double r40453488 = v;
        double r40453489 = r40453488 * r40453488;
        double r40453490 = r40453487 * r40453489;
        double r40453491 = 2.0;
        double r40453492 = r40453490 + r40453491;
        double r40453493 = sqrt(r40453492);
        double r40453494 = r40453493 * r40453489;
        double r40453495 = r40453493 - r40453494;
        double r40453496 = r40453484 / r40453495;
        double r40453497 = r40453486 * r40453496;
        return r40453497;
}

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

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

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

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

  6. Applied distribute-rgt-out--0.0

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

  7. Applied add-sqr-sqrt0.0

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

  8. Applied times-frac0.0

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

  9. Applied times-frac0.0

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

  10. Simplified0.0

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

  11. Simplified0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2019120 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))