Average Error: 0.0 → 0.0
Time: 13.0s
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 r154573 = 2.0;
        double r154574 = sqrt(r154573);
        double r154575 = 4.0;
        double r154576 = r154574 / r154575;
        double r154577 = 1.0;
        double r154578 = 3.0;
        double r154579 = v;
        double r154580 = r154579 * r154579;
        double r154581 = r154578 * r154580;
        double r154582 = r154577 - r154581;
        double r154583 = sqrt(r154582);
        double r154584 = r154576 * r154583;
        double r154585 = r154577 - r154580;
        double r154586 = r154584 * r154585;
        return r154586;
}
↓
double f(double v) {
        double r154587 = 2.0;
        double r154588 = sqrt(r154587);
        double r154589 = 4.0;
        double r154590 = r154588 / r154589;
        double r154591 = 1.0;
        double r154592 = 3.0;
        double r154593 = v;
        double r154594 = r154593 * r154593;
        double r154595 = r154592 * r154594;
        double r154596 = r154591 - r154595;
        double r154597 = cbrt(r154596);
        double r154598 = fabs(r154597);
        double r154599 = sqrt(r154597);
        double r154600 = r154598 * r154599;
        double r154601 = r154590 * r154600;
        double r154602 = r154591 - r154594;
        double r154603 = r154601 * r154602;
        return r154603;
}

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 2019303 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))