Average Error: 0.0 → 0.0
Time: 8.3s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\left(\left({1}^{3} - {\left(v \cdot v\right)}^{3}\right) \cdot {\left(\sqrt{2}\right)}^{\frac{2}{3}}\right) \cdot \left(\frac{\sqrt[3]{\sqrt{2}}}{\sqrt{4}} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)}{\left({v}^{2} \cdot \left({v}^{2} + 1\right) + 1 \cdot 1\right) \cdot \sqrt{4}}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{\left(\left({1}^{3} - {\left(v \cdot v\right)}^{3}\right) \cdot {\left(\sqrt{2}\right)}^{\frac{2}{3}}\right) \cdot \left(\frac{\sqrt[3]{\sqrt{2}}}{\sqrt{4}} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)}{\left({v}^{2} \cdot \left({v}^{2} + 1\right) + 1 \cdot 1\right) \cdot \sqrt{4}}
double f(double v) {
        double r266407 = 2.0;
        double r266408 = sqrt(r266407);
        double r266409 = 4.0;
        double r266410 = r266408 / r266409;
        double r266411 = 1.0;
        double r266412 = 3.0;
        double r266413 = v;
        double r266414 = r266413 * r266413;
        double r266415 = r266412 * r266414;
        double r266416 = r266411 - r266415;
        double r266417 = sqrt(r266416);
        double r266418 = r266410 * r266417;
        double r266419 = r266411 - r266414;
        double r266420 = r266418 * r266419;
        return r266420;
}
↓
double f(double v) {
        double r266421 = 1.0;
        double r266422 = 3.0;
        double r266423 = pow(r266421, r266422);
        double r266424 = v;
        double r266425 = r266424 * r266424;
        double r266426 = pow(r266425, r266422);
        double r266427 = r266423 - r266426;
        double r266428 = 2.0;
        double r266429 = sqrt(r266428);
        double r266430 = 0.6666666666666666;
        double r266431 = pow(r266429, r266430);
        double r266432 = r266427 * r266431;
        double r266433 = cbrt(r266429);
        double r266434 = 4.0;
        double r266435 = sqrt(r266434);
        double r266436 = r266433 / r266435;
        double r266437 = 3.0;
        double r266438 = r266437 * r266425;
        double r266439 = r266421 - r266438;
        double r266440 = sqrt(r266439);
        double r266441 = r266436 * r266440;
        double r266442 = r266432 * r266441;
        double r266443 = 2.0;
        double r266444 = pow(r266424, r266443);
        double r266445 = r266444 + r266421;
        double r266446 = r266444 * r266445;
        double r266447 = r266421 * r266421;
        double r266448 = r266446 + r266447;
        double r266449 = r266448 * r266435;
        double r266450 = r266442 / r266449;
        return r266450;
}

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. Using strategy rm

  8. Applied flip3--0.0

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

  9. Applied associate-*l/0.0

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

  10. Applied frac-times0.0

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

  11. Simplified0.0

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

  12. Simplified0.0

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

  13. Final simplification0.0

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

Reproduce

herbie shell --seed 2020056 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))