Average Error: 0.0 → 0.0
Time: 6.2s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(1 - v \cdot v\right) \cdot \left(\left(\frac{\sqrt[3]{\sqrt{2}}}{4} \cdot \left(\sqrt{\sqrt{1 - 3 \cdot \left(v \cdot v\right)}} \cdot \sqrt{\sqrt{1 - 3 \cdot \left(v \cdot v\right)}}\right)\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)\]
\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 \left(\sqrt{\sqrt{1 - 3 \cdot \left(v \cdot v\right)}} \cdot \sqrt{\sqrt{1 - 3 \cdot \left(v \cdot v\right)}}\right)\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)
double f(double v) {
        double r282542 = 2.0;
        double r282543 = sqrt(r282542);
        double r282544 = 4.0;
        double r282545 = r282543 / r282544;
        double r282546 = 1.0;
        double r282547 = 3.0;
        double r282548 = v;
        double r282549 = r282548 * r282548;
        double r282550 = r282547 * r282549;
        double r282551 = r282546 - r282550;
        double r282552 = sqrt(r282551);
        double r282553 = r282545 * r282552;
        double r282554 = r282546 - r282549;
        double r282555 = r282553 * r282554;
        return r282555;
}
↓
double f(double v) {
        double r282556 = 1.0;
        double r282557 = v;
        double r282558 = r282557 * r282557;
        double r282559 = r282556 - r282558;
        double r282560 = 2.0;
        double r282561 = sqrt(r282560);
        double r282562 = cbrt(r282561);
        double r282563 = 4.0;
        double r282564 = r282562 / r282563;
        double r282565 = 3.0;
        double r282566 = r282565 * r282558;
        double r282567 = r282556 - r282566;
        double r282568 = sqrt(r282567);
        double r282569 = sqrt(r282568);
        double r282570 = r282569 * r282569;
        double r282571 = r282564 * r282570;
        double r282572 = r282562 * r282562;
        double r282573 = r282571 * r282572;
        double r282574 = r282559 * r282573;
        return r282574;
}

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. Using strategy rm

  8. Applied add-sqr-sqrt0.0

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

  9. Applied sqrt-prod0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019322 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))