Average Error: 0.0 → 0.0
Time: 12.6s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\sqrt[3]{{\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)}^{3}} \cdot \left(1 - v \cdot v\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\sqrt[3]{{\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)}^{3}} \cdot \left(1 - v \cdot v\right)
double f(double v) {
        double r164149 = 2.0;
        double r164150 = sqrt(r164149);
        double r164151 = 4.0;
        double r164152 = r164150 / r164151;
        double r164153 = 1.0;
        double r164154 = 3.0;
        double r164155 = v;
        double r164156 = r164155 * r164155;
        double r164157 = r164154 * r164156;
        double r164158 = r164153 - r164157;
        double r164159 = sqrt(r164158);
        double r164160 = r164152 * r164159;
        double r164161 = r164153 - r164156;
        double r164162 = r164160 * r164161;
        return r164162;
}
↓
double f(double v) {
        double r164163 = 2.0;
        double r164164 = sqrt(r164163);
        double r164165 = 4.0;
        double r164166 = r164164 / r164165;
        double r164167 = 1.0;
        double r164168 = 3.0;
        double r164169 = v;
        double r164170 = r164169 * r164169;
        double r164171 = r164168 * r164170;
        double r164172 = r164167 - r164171;
        double r164173 = sqrt(r164172);
        double r164174 = r164166 * r164173;
        double r164175 = 3.0;
        double r164176 = pow(r164174, r164175);
        double r164177 = cbrt(r164176);
        double r164178 = r164167 - r164170;
        double r164179 = r164177 * r164178;
        return r164179;
}

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-cbrt-cube0.0

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

  4. Applied add-cbrt-cube0.0

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

  5. Applied add-cbrt-cube1.0

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

  6. Applied cbrt-undiv0.0

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

  7. Applied cbrt-unprod0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019209 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))