Average Error: 0.0 → 0.0
Time: 3.6s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(1 - v \cdot v\right) \cdot \left(\left(\frac{\sqrt[3]{\sqrt{2}}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\left(1 - v \cdot v\right) \cdot \left(\left(\frac{\sqrt[3]{\sqrt{2}}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)
double f(double v) {
        double r328294 = 2.0;
        double r328295 = sqrt(r328294);
        double r328296 = 4.0;
        double r328297 = r328295 / r328296;
        double r328298 = 1.0;
        double r328299 = 3.0;
        double r328300 = v;
        double r328301 = r328300 * r328300;
        double r328302 = r328299 * r328301;
        double r328303 = r328298 - r328302;
        double r328304 = sqrt(r328303);
        double r328305 = r328297 * r328304;
        double r328306 = r328298 - r328301;
        double r328307 = r328305 * r328306;
        return r328307;
}
↓
double f(double v) {
        double r328308 = 1.0;
        double r328309 = v;
        double r328310 = r328309 * r328309;
        double r328311 = r328308 - r328310;
        double r328312 = 2.0;
        double r328313 = sqrt(r328312);
        double r328314 = cbrt(r328313);
        double r328315 = 4.0;
        double r328316 = r328314 / r328315;
        double r328317 = 3.0;
        double r328318 = r328317 * r328310;
        double r328319 = r328308 - r328318;
        double r328320 = sqrt(r328319);
        double r328321 = r328316 * r328320;
        double r328322 = r328314 * r328314;
        double r328323 = r328321 * r328322;
        double r328324 = r328311 * r328323;
        return r328324;
}

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 *-un-lft-identity0.0

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

  4. Applied add-cube-cbrt0.0

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

  5. Applied times-frac0.0

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

  6. Applied associate-*l*0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019362 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))