Average Error: 0.0 → 0.0
Time: 14.2s
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 r194766 = 2.0;
        double r194767 = sqrt(r194766);
        double r194768 = 4.0;
        double r194769 = r194767 / r194768;
        double r194770 = 1.0;
        double r194771 = 3.0;
        double r194772 = v;
        double r194773 = r194772 * r194772;
        double r194774 = r194771 * r194773;
        double r194775 = r194770 - r194774;
        double r194776 = sqrt(r194775);
        double r194777 = r194769 * r194776;
        double r194778 = r194770 - r194773;
        double r194779 = r194777 * r194778;
        return r194779;
}
↓
double f(double v) {
        double r194780 = 2.0;
        double r194781 = sqrt(r194780);
        double r194782 = 4.0;
        double r194783 = r194781 / r194782;
        double r194784 = 1.0;
        double r194785 = 3.0;
        double r194786 = v;
        double r194787 = r194786 * r194786;
        double r194788 = r194785 * r194787;
        double r194789 = r194784 - r194788;
        double r194790 = cbrt(r194789);
        double r194791 = fabs(r194790);
        double r194792 = sqrt(r194790);
        double r194793 = r194791 * r194792;
        double r194794 = r194783 * r194793;
        double r194795 = r194784 - r194787;
        double r194796 = r194794 * r194795;
        return r194796;
}

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 +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))))