Average Error: 0.0 → 0.0
Time: 9.8s
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 r151436 = 2.0;
        double r151437 = sqrt(r151436);
        double r151438 = 4.0;
        double r151439 = r151437 / r151438;
        double r151440 = 1.0;
        double r151441 = 3.0;
        double r151442 = v;
        double r151443 = r151442 * r151442;
        double r151444 = r151441 * r151443;
        double r151445 = r151440 - r151444;
        double r151446 = sqrt(r151445);
        double r151447 = r151439 * r151446;
        double r151448 = r151440 - r151443;
        double r151449 = r151447 * r151448;
        return r151449;
}
↓
double f(double v) {
        double r151450 = 2.0;
        double r151451 = sqrt(r151450);
        double r151452 = cbrt(r151451);
        double r151453 = r151452 * r151452;
        double r151454 = 4.0;
        double r151455 = sqrt(r151454);
        double r151456 = r151453 / r151455;
        double r151457 = r151452 / r151455;
        double r151458 = 1.0;
        double r151459 = 3.0;
        double r151460 = v;
        double r151461 = r151460 * r151460;
        double r151462 = r151459 * r151461;
        double r151463 = r151458 - r151462;
        double r151464 = sqrt(r151463);
        double r151465 = r151457 * r151464;
        double r151466 = r151456 * r151465;
        double r151467 = r151458 - r151461;
        double r151468 = r151466 * r151467;
        return r151468;
}

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))))