Average Error: 0.0 → 0.0
Time: 3.2s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(\frac{\sqrt{2} \cdot \left|\sqrt[3]{1 - 3 \cdot \left(v \cdot v\right)}\right|}{4} \cdot \sqrt{\sqrt[3]{1 - 3 \cdot \left(v \cdot v\right)}}\right) \cdot \left(1 - v \cdot v\right)\]
\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 r269804 = 2.0;
        double r269805 = sqrt(r269804);
        double r269806 = 4.0;
        double r269807 = r269805 / r269806;
        double r269808 = 1.0;
        double r269809 = 3.0;
        double r269810 = v;
        double r269811 = r269810 * r269810;
        double r269812 = r269809 * r269811;
        double r269813 = r269808 - r269812;
        double r269814 = sqrt(r269813);
        double r269815 = r269807 * r269814;
        double r269816 = r269808 - r269811;
        double r269817 = r269815 * r269816;
        return r269817;
}
↓
double f(double v) {
        double r269818 = 2.0;
        double r269819 = sqrt(r269818);
        double r269820 = 1.0;
        double r269821 = 3.0;
        double r269822 = v;
        double r269823 = r269822 * r269822;
        double r269824 = r269821 * r269823;
        double r269825 = r269820 - r269824;
        double r269826 = cbrt(r269825);
        double r269827 = fabs(r269826);
        double r269828 = r269819 * r269827;
        double r269829 = 4.0;
        double r269830 = r269828 / r269829;
        double r269831 = sqrt(r269826);
        double r269832 = r269830 * r269831;
        double r269833 = r269820 - r269823;
        double r269834 = r269832 * r269833;
        return r269834;
}

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