Average Error: 1.0 → 0.0
Time: 23.1s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{4}}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)} \cdot \left(\frac{\left|\sqrt[3]{4}\right|}{\left|\sqrt[3]{2 - 6 \cdot \left(v \cdot v\right)}\right|} \cdot \frac{\sqrt{\sqrt[3]{4}}}{\sqrt{\sqrt[3]{2 - 6 \cdot \left(v \cdot v\right)}}}\right)\right)\right)\]
\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)}}
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{4}}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)} \cdot \left(\frac{\left|\sqrt[3]{4}\right|}{\left|\sqrt[3]{2 - 6 \cdot \left(v \cdot v\right)}\right|} \cdot \frac{\sqrt{\sqrt[3]{4}}}{\sqrt{\sqrt[3]{2 - 6 \cdot \left(v \cdot v\right)}}}\right)\right)\right)
double f(double v) {
        double r221587 = 4.0;
        double r221588 = 3.0;
        double r221589 = atan2(1.0, 0.0);
        double r221590 = r221588 * r221589;
        double r221591 = 1.0;
        double r221592 = v;
        double r221593 = r221592 * r221592;
        double r221594 = r221591 - r221593;
        double r221595 = r221590 * r221594;
        double r221596 = 2.0;
        double r221597 = 6.0;
        double r221598 = r221597 * r221593;
        double r221599 = r221596 - r221598;
        double r221600 = sqrt(r221599);
        double r221601 = r221595 * r221600;
        double r221602 = r221587 / r221601;
        return r221602;
}
↓
double f(double v) {
        double r221603 = 4.0;
        double r221604 = sqrt(r221603);
        double r221605 = 3.0;
        double r221606 = atan2(1.0, 0.0);
        double r221607 = r221605 * r221606;
        double r221608 = 1.0;
        double r221609 = v;
        double r221610 = r221609 * r221609;
        double r221611 = r221608 - r221610;
        double r221612 = r221607 * r221611;
        double r221613 = r221604 / r221612;
        double r221614 = cbrt(r221603);
        double r221615 = fabs(r221614);
        double r221616 = 2.0;
        double r221617 = 6.0;
        double r221618 = r221617 * r221610;
        double r221619 = r221616 - r221618;
        double r221620 = cbrt(r221619);
        double r221621 = fabs(r221620);
        double r221622 = r221615 / r221621;
        double r221623 = sqrt(r221614);
        double r221624 = sqrt(r221620);
        double r221625 = r221623 / r221624;
        double r221626 = r221622 * r221625;
        double r221627 = r221613 * r221626;
        double r221628 = log1p(r221627);
        double r221629 = expm1(r221628);
        return r221629;
}

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-cube-cbrt1.0

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

  7. Applied sqrt-prod1.0

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

  8. Applied add-cube-cbrt1.0

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

  9. Applied sqrt-prod1.0

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

  10. Applied times-frac0.0

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

  11. Simplified0.0

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

  13. Applied expm1-log1p-u0.0

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{4}}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)} \cdot \left(\frac{\left|\sqrt[3]{4}\right|}{\left|\sqrt[3]{2 - 6 \cdot \left(v \cdot v\right)}\right|} \cdot \frac{\sqrt{\sqrt[3]{4}}}{\sqrt{\sqrt[3]{2 - 6 \cdot \left(v \cdot v\right)}}}\right)\right)\right)}\]

  14. Final simplification0.0

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{4}}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)} \cdot \left(\frac{\left|\sqrt[3]{4}\right|}{\left|\sqrt[3]{2 - 6 \cdot \left(v \cdot v\right)}\right|} \cdot \frac{\sqrt{\sqrt[3]{4}}}{\sqrt{\sqrt[3]{2 - 6 \cdot \left(v \cdot v\right)}}}\right)\right)\right)\]

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))