Average Error: 0.0 → 0.0
Time: 6.1s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(1 - v \cdot v\right) \cdot \left(\left(\left({\left(\sqrt{2}\right)}^{\frac{1}{3}} \cdot \frac{\sqrt[3]{\sqrt{2}}}{4}\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \sqrt[3]{\sqrt{2}}\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\left(1 - v \cdot v\right) \cdot \left(\left(\left({\left(\sqrt{2}\right)}^{\frac{1}{3}} \cdot \frac{\sqrt[3]{\sqrt{2}}}{4}\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \sqrt[3]{\sqrt{2}}\right)
double f(double v) {
        double r265602 = 2.0;
        double r265603 = sqrt(r265602);
        double r265604 = 4.0;
        double r265605 = r265603 / r265604;
        double r265606 = 1.0;
        double r265607 = 3.0;
        double r265608 = v;
        double r265609 = r265608 * r265608;
        double r265610 = r265607 * r265609;
        double r265611 = r265606 - r265610;
        double r265612 = sqrt(r265611);
        double r265613 = r265605 * r265612;
        double r265614 = r265606 - r265609;
        double r265615 = r265613 * r265614;
        return r265615;
}
↓
double f(double v) {
        double r265616 = 1.0;
        double r265617 = v;
        double r265618 = r265617 * r265617;
        double r265619 = r265616 - r265618;
        double r265620 = 2.0;
        double r265621 = sqrt(r265620);
        double r265622 = 0.3333333333333333;
        double r265623 = pow(r265621, r265622);
        double r265624 = cbrt(r265621);
        double r265625 = 4.0;
        double r265626 = r265624 / r265625;
        double r265627 = r265623 * r265626;
        double r265628 = 3.0;
        double r265629 = r265628 * r265618;
        double r265630 = r265616 - r265629;
        double r265631 = sqrt(r265630);
        double r265632 = r265627 * r265631;
        double r265633 = r265632 * r265624;
        double r265634 = r265619 * r265633;
        return r265634;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
  2. Using strategy rm

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

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

  4. Applied add-cube-cbrt0.0

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

  5. Applied times-frac0.0

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

  6. Applied associate-*l*0.0

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

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

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

  9. Applied times-frac0.0

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

  10. Applied associate-*l*0.0

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

  11. Simplified0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))