Average Error: 0.0 → 0.0
Time: 4.5s
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 r153305 = 2.0;
        double r153306 = sqrt(r153305);
        double r153307 = 4.0;
        double r153308 = r153306 / r153307;
        double r153309 = 1.0;
        double r153310 = 3.0;
        double r153311 = v;
        double r153312 = r153311 * r153311;
        double r153313 = r153310 * r153312;
        double r153314 = r153309 - r153313;
        double r153315 = sqrt(r153314);
        double r153316 = r153308 * r153315;
        double r153317 = r153309 - r153312;
        double r153318 = r153316 * r153317;
        return r153318;
}
↓
double f(double v) {
        double r153319 = 2.0;
        double r153320 = sqrt(r153319);
        double r153321 = cbrt(r153320);
        double r153322 = r153321 * r153321;
        double r153323 = 4.0;
        double r153324 = sqrt(r153323);
        double r153325 = r153322 / r153324;
        double r153326 = r153321 / r153324;
        double r153327 = 1.0;
        double r153328 = 3.0;
        double r153329 = v;
        double r153330 = r153329 * r153329;
        double r153331 = r153328 * r153330;
        double r153332 = r153327 - r153331;
        double r153333 = sqrt(r153332);
        double r153334 = r153326 * r153333;
        double r153335 = r153325 * r153334;
        double r153336 = r153327 - r153330;
        double r153337 = r153335 * r153336;
        return r153337;
}

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 2020035 +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))))