Average Error: 0.0 → 0.0
Time: 10.0s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(\frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\sqrt{4}} \cdot \left(\frac{\sqrt[3]{\sqrt{2}}}{\sqrt{4}} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)\right) \cdot \left(1 - v \cdot v\right)\]
\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[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\sqrt{4}} \cdot \left(\frac{\sqrt[3]{\sqrt{2}}}{\sqrt{4}} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)\right) \cdot \left(1 - v \cdot v\right)
double f(double v) {
        double r261306 = 2.0;
        double r261307 = sqrt(r261306);
        double r261308 = 4.0;
        double r261309 = r261307 / r261308;
        double r261310 = 1.0;
        double r261311 = 3.0;
        double r261312 = v;
        double r261313 = r261312 * r261312;
        double r261314 = r261311 * r261313;
        double r261315 = r261310 - r261314;
        double r261316 = sqrt(r261315);
        double r261317 = r261309 * r261316;
        double r261318 = r261310 - r261313;
        double r261319 = r261317 * r261318;
        return r261319;
}
↓
double f(double v) {
        double r261320 = 2.0;
        double r261321 = sqrt(r261320);
        double r261322 = cbrt(r261321);
        double r261323 = r261322 * r261322;
        double r261324 = 4.0;
        double r261325 = sqrt(r261324);
        double r261326 = r261323 / r261325;
        double r261327 = r261322 / r261325;
        double r261328 = 1.0;
        double r261329 = 3.0;
        double r261330 = v;
        double r261331 = r261330 * r261330;
        double r261332 = r261329 * r261331;
        double r261333 = r261328 - r261332;
        double r261334 = sqrt(r261333);
        double r261335 = r261327 * r261334;
        double r261336 = r261326 * r261335;
        double r261337 = r261328 - r261331;
        double r261338 = r261336 * r261337;
        return r261338;
}

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-sqr-sqrt0.0

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

  4. Applied add-cube-cbrt0.0

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

  5. Applied times-frac0.0

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

  6. Applied associate-*l*0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020045 +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))))