Average Error: 1.0 → 0.0
Time: 19.7s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[e^{\log \left(\frac{\sqrt{4}}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}\right)} \cdot \frac{\sqrt{4}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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)}}
↓
e^{\log \left(\frac{\sqrt{4}}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}\right)} \cdot \frac{\sqrt{4}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r177555 = 4.0;
        double r177556 = 3.0;
        double r177557 = atan2(1.0, 0.0);
        double r177558 = r177556 * r177557;
        double r177559 = 1.0;
        double r177560 = v;
        double r177561 = r177560 * r177560;
        double r177562 = r177559 - r177561;
        double r177563 = r177558 * r177562;
        double r177564 = 2.0;
        double r177565 = 6.0;
        double r177566 = r177565 * r177561;
        double r177567 = r177564 - r177566;
        double r177568 = sqrt(r177567);
        double r177569 = r177563 * r177568;
        double r177570 = r177555 / r177569;
        return r177570;
}
↓
double f(double v) {
        double r177571 = 4.0;
        double r177572 = sqrt(r177571);
        double r177573 = 3.0;
        double r177574 = atan2(1.0, 0.0);
        double r177575 = r177573 * r177574;
        double r177576 = 1.0;
        double r177577 = v;
        double r177578 = r177577 * r177577;
        double r177579 = r177576 - r177578;
        double r177580 = r177575 * r177579;
        double r177581 = r177572 / r177580;
        double r177582 = log(r177581);
        double r177583 = exp(r177582);
        double r177584 = 2.0;
        double r177585 = 6.0;
        double r177586 = r177585 * r177578;
        double r177587 = r177584 - r177586;
        double r177588 = sqrt(r177587);
        double r177589 = r177572 / r177588;
        double r177590 = r177583 * r177589;
        return r177590;
}

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. Using strategy rm

  3. Applied add-sqr-sqrt1.0

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

  4. Applied times-frac0.0

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

  6. Applied add-exp-log0.0

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

  7. Applied add-exp-log0.0

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

  8. Applied add-exp-log1.0

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

  9. Applied prod-exp1.0

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

  10. Applied prod-exp1.0

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

  11. Applied add-exp-log1.0

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

  12. Applied div-exp0.0

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

  13. Simplified0.0

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

  14. Final simplification0.0

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

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))