Average Error: 0.0 → 0.0
Time: 16.2s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \sqrt{1 \cdot 1 - {v}^{4}}}{\sqrt{1 + v \cdot v}} \cdot \sqrt{1 - v \cdot v}\]
\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(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \sqrt{1 \cdot 1 - {v}^{4}}}{\sqrt{1 + v \cdot v}} \cdot \sqrt{1 - v \cdot v}
double f(double v) {
        double r158481 = 2.0;
        double r158482 = sqrt(r158481);
        double r158483 = 4.0;
        double r158484 = r158482 / r158483;
        double r158485 = 1.0;
        double r158486 = 3.0;
        double r158487 = v;
        double r158488 = r158487 * r158487;
        double r158489 = r158486 * r158488;
        double r158490 = r158485 - r158489;
        double r158491 = sqrt(r158490);
        double r158492 = r158484 * r158491;
        double r158493 = r158485 - r158488;
        double r158494 = r158492 * r158493;
        return r158494;
}
↓
double f(double v) {
        double r158495 = 2.0;
        double r158496 = sqrt(r158495);
        double r158497 = 4.0;
        double r158498 = r158496 / r158497;
        double r158499 = 1.0;
        double r158500 = 3.0;
        double r158501 = v;
        double r158502 = r158501 * r158501;
        double r158503 = r158500 * r158502;
        double r158504 = r158499 - r158503;
        double r158505 = sqrt(r158504);
        double r158506 = r158498 * r158505;
        double r158507 = r158499 * r158499;
        double r158508 = 4.0;
        double r158509 = pow(r158501, r158508);
        double r158510 = r158507 - r158509;
        double r158511 = sqrt(r158510);
        double r158512 = r158506 * r158511;
        double r158513 = r158499 + r158502;
        double r158514 = sqrt(r158513);
        double r158515 = r158512 / r158514;
        double r158516 = r158499 - r158502;
        double r158517 = sqrt(r158516);
        double r158518 = r158515 * r158517;
        return r158518;
}

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

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

  4. Applied associate-*r*0.0

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

  6. Applied flip--0.0

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

  7. Applied sqrt-div0.0

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

  8. Applied associate-*r/0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019325 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))