Average Error: 0.0 → 0.0
Time: 16.6s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\sqrt[3]{\frac{\left(\left(\left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \sqrt{2}\right)\right) \cdot \left(\left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \sqrt{2}\right)\right)}{\left(\left(4 \cdot \left(1 + v \cdot v\right)\right) \cdot \left(4 \cdot \left(1 + v \cdot v\right)\right)\right) \cdot \left(4 \cdot \left(1 + v \cdot v\right)\right)}}\]
\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]{\frac{\left(\left(\left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \sqrt{2}\right)\right) \cdot \left(\left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \sqrt{2}\right)\right)}{\left(\left(4 \cdot \left(1 + v \cdot v\right)\right) \cdot \left(4 \cdot \left(1 + v \cdot v\right)\right)\right) \cdot \left(4 \cdot \left(1 + v \cdot v\right)\right)}}
double f(double v) {
        double r16348146 = 2.0;
        double r16348147 = sqrt(r16348146);
        double r16348148 = 4.0;
        double r16348149 = r16348147 / r16348148;
        double r16348150 = 1.0;
        double r16348151 = 3.0;
        double r16348152 = v;
        double r16348153 = r16348152 * r16348152;
        double r16348154 = r16348151 * r16348153;
        double r16348155 = r16348150 - r16348154;
        double r16348156 = sqrt(r16348155);
        double r16348157 = r16348149 * r16348156;
        double r16348158 = r16348150 - r16348153;
        double r16348159 = r16348157 * r16348158;
        return r16348159;
}
↓
double f(double v) {
        double r16348160 = 1.0;
        double r16348161 = v;
        double r16348162 = r16348161 * r16348161;
        double r16348163 = r16348162 * r16348162;
        double r16348164 = r16348160 - r16348163;
        double r16348165 = 3.0;
        double r16348166 = r16348162 * r16348165;
        double r16348167 = r16348160 - r16348166;
        double r16348168 = sqrt(r16348167);
        double r16348169 = 2.0;
        double r16348170 = sqrt(r16348169);
        double r16348171 = r16348168 * r16348170;
        double r16348172 = r16348164 * r16348171;
        double r16348173 = r16348172 * r16348172;
        double r16348174 = r16348173 * r16348172;
        double r16348175 = 4.0;
        double r16348176 = r16348160 + r16348162;
        double r16348177 = r16348175 * r16348176;
        double r16348178 = r16348177 * r16348177;
        double r16348179 = r16348178 * r16348177;
        double r16348180 = r16348174 / r16348179;
        double r16348181 = cbrt(r16348180);
        return r16348181;
}

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 flip--0.0

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

  4. Applied associate-*l/0.0

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

  5. Applied frac-times0.0

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

  6. Simplified0.0

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

  8. Applied add-cbrt-cube0.0

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

  9. Applied add-cbrt-cube1.0

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

  10. Applied cbrt-undiv0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019162 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))