Average Error: 0.0 → 0.0
Time: 4.4s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(\frac{\sqrt{2} \cdot \left|\sqrt[3]{1 - 3 \cdot \left(v \cdot v\right)}\right|}{4} \cdot \sqrt{\sqrt[3]{1 - 3 \cdot \left(v \cdot v\right)}}\right) \cdot \left(1 - v \cdot v\right)\]
\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} \cdot \left|\sqrt[3]{1 - 3 \cdot \left(v \cdot v\right)}\right|}{4} \cdot \sqrt{\sqrt[3]{1 - 3 \cdot \left(v \cdot v\right)}}\right) \cdot \left(1 - v \cdot v\right)
double f(double v) {
        double r431616 = 2.0;
        double r431617 = sqrt(r431616);
        double r431618 = 4.0;
        double r431619 = r431617 / r431618;
        double r431620 = 1.0;
        double r431621 = 3.0;
        double r431622 = v;
        double r431623 = r431622 * r431622;
        double r431624 = r431621 * r431623;
        double r431625 = r431620 - r431624;
        double r431626 = sqrt(r431625);
        double r431627 = r431619 * r431626;
        double r431628 = r431620 - r431623;
        double r431629 = r431627 * r431628;
        return r431629;
}
↓
double f(double v) {
        double r431630 = 2.0;
        double r431631 = sqrt(r431630);
        double r431632 = 1.0;
        double r431633 = 3.0;
        double r431634 = v;
        double r431635 = r431634 * r431634;
        double r431636 = r431633 * r431635;
        double r431637 = r431632 - r431636;
        double r431638 = cbrt(r431637);
        double r431639 = fabs(r431638);
        double r431640 = r431631 * r431639;
        double r431641 = 4.0;
        double r431642 = r431640 / r431641;
        double r431643 = sqrt(r431638);
        double r431644 = r431642 * r431643;
        double r431645 = r431632 - r431635;
        double r431646 = r431644 * r431645;
        return r431646;
}

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. Applied associate-*r*0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020059 +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))))