Average Error: 0.0 → 0.0
Time: 25.6s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\sqrt[3]{{\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \frac{\sqrt{2}}{4}\right)}^{3} \cdot {\left({\left(1 - v \cdot v\right)}^{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right)}\right)}^{\left(\sqrt[3]{3}\right)}}\]
\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(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \frac{\sqrt{2}}{4}\right)}^{3} \cdot {\left({\left(1 - v \cdot v\right)}^{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right)}\right)}^{\left(\sqrt[3]{3}\right)}}
double f(double v) {
        double r269115 = 2.0;
        double r269116 = sqrt(r269115);
        double r269117 = 4.0;
        double r269118 = r269116 / r269117;
        double r269119 = 1.0;
        double r269120 = 3.0;
        double r269121 = v;
        double r269122 = r269121 * r269121;
        double r269123 = r269120 * r269122;
        double r269124 = r269119 - r269123;
        double r269125 = sqrt(r269124);
        double r269126 = r269118 * r269125;
        double r269127 = r269119 - r269122;
        double r269128 = r269126 * r269127;
        return r269128;
}
↓
double f(double v) {
        double r269129 = 1.0;
        double r269130 = 3.0;
        double r269131 = v;
        double r269132 = r269131 * r269131;
        double r269133 = r269130 * r269132;
        double r269134 = r269129 - r269133;
        double r269135 = sqrt(r269134);
        double r269136 = 2.0;
        double r269137 = sqrt(r269136);
        double r269138 = 4.0;
        double r269139 = r269137 / r269138;
        double r269140 = r269135 * r269139;
        double r269141 = 3.0;
        double r269142 = pow(r269140, r269141);
        double r269143 = r269129 - r269132;
        double r269144 = cbrt(r269141);
        double r269145 = r269144 * r269144;
        double r269146 = pow(r269143, r269145);
        double r269147 = pow(r269146, r269144);
        double r269148 = r269142 * r269147;
        double r269149 = cbrt(r269148);
        return r269149;
}

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

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

  4. Applied add-cbrt-cube0.0

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

  5. Applied add-cbrt-cube0.0

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

  6. Applied add-cbrt-cube1.0

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

  7. Applied cbrt-undiv0.0

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

  8. Applied cbrt-unprod0.0

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

  9. Applied cbrt-unprod0.0

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

  10. Simplified0.0

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

  12. Applied unpow-prod-down0.0

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

  13. Simplified0.0

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

  15. Applied add-cube-cbrt0.0

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

  16. Applied pow-unpow0.0

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

  17. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 +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))))