Average Error: 0.0 → 0.0
Time: 4.9s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(\frac{\sqrt{2}}{4} \cdot \left(\left|\sqrt[3]{1 - 3 \cdot \left(v \cdot v\right)}\right| \cdot \sqrt{\sqrt[3]{1 - 3 \cdot \left(v \cdot v\right)}}\right)\right) \cdot \left(1 - v \cdot v\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\left(\frac{\sqrt{2}}{4} \cdot \left(\left|\sqrt[3]{1 - 3 \cdot \left(v \cdot v\right)}\right| \cdot \sqrt{\sqrt[3]{1 - 3 \cdot \left(v \cdot v\right)}}\right)\right) \cdot \left(1 - v \cdot v\right)
double f(double v) {
        double r358614 = 2.0;
        double r358615 = sqrt(r358614);
        double r358616 = 4.0;
        double r358617 = r358615 / r358616;
        double r358618 = 1.0;
        double r358619 = 3.0;
        double r358620 = v;
        double r358621 = r358620 * r358620;
        double r358622 = r358619 * r358621;
        double r358623 = r358618 - r358622;
        double r358624 = sqrt(r358623);
        double r358625 = r358617 * r358624;
        double r358626 = r358618 - r358621;
        double r358627 = r358625 * r358626;
        return r358627;
}
↓
double f(double v) {
        double r358628 = 2.0;
        double r358629 = sqrt(r358628);
        double r358630 = 4.0;
        double r358631 = r358629 / r358630;
        double r358632 = 1.0;
        double r358633 = 3.0;
        double r358634 = v;
        double r358635 = r358634 * r358634;
        double r358636 = r358633 * r358635;
        double r358637 = r358632 - r358636;
        double r358638 = cbrt(r358637);
        double r358639 = fabs(r358638);
        double r358640 = sqrt(r358638);
        double r358641 = r358639 * r358640;
        double r358642 = r358631 * r358641;
        double r358643 = r358632 - r358635;
        double r358644 = r358642 * r358643;
        return r358644;
}

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 add-cube-cbrt0.0

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

  4. Applied sqrt-prod0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020046 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))