Average Error: 0.0 → 0.0
Time: 30.9s
Precision: 64
\[\left(\frac{\sqrt{2.0}}{4.0} \cdot \sqrt{1.0 - 3.0 \cdot \left(v \cdot v\right)}\right) \cdot \left(1.0 - v \cdot v\right)\]
↓
\[\frac{\left(\left(\left(\left(1.0 - v \cdot v\right) \cdot \sqrt[3]{\sqrt{2.0}}\right) \cdot \sqrt[3]{\sqrt{2.0}}\right) \cdot \frac{\sqrt[3]{\sqrt{2.0}}}{4.0}\right) \cdot \sqrt{\left(1.0 - \left(v \cdot v\right) \cdot 3.0\right) \cdot \left(\left(v \cdot v\right) \cdot 3.0 + 1.0\right)}}{\sqrt{\left(v \cdot v\right) \cdot 3.0 + 1.0}}\]
\left(\frac{\sqrt{2.0}}{4.0} \cdot \sqrt{1.0 - 3.0 \cdot \left(v \cdot v\right)}\right) \cdot \left(1.0 - v \cdot v\right)
↓
\frac{\left(\left(\left(\left(1.0 - v \cdot v\right) \cdot \sqrt[3]{\sqrt{2.0}}\right) \cdot \sqrt[3]{\sqrt{2.0}}\right) \cdot \frac{\sqrt[3]{\sqrt{2.0}}}{4.0}\right) \cdot \sqrt{\left(1.0 - \left(v \cdot v\right) \cdot 3.0\right) \cdot \left(\left(v \cdot v\right) \cdot 3.0 + 1.0\right)}}{\sqrt{\left(v \cdot v\right) \cdot 3.0 + 1.0}}
double f(double v) {
        double r6865608 = 2.0;
        double r6865609 = sqrt(r6865608);
        double r6865610 = 4.0;
        double r6865611 = r6865609 / r6865610;
        double r6865612 = 1.0;
        double r6865613 = 3.0;
        double r6865614 = v;
        double r6865615 = r6865614 * r6865614;
        double r6865616 = r6865613 * r6865615;
        double r6865617 = r6865612 - r6865616;
        double r6865618 = sqrt(r6865617);
        double r6865619 = r6865611 * r6865618;
        double r6865620 = r6865612 - r6865615;
        double r6865621 = r6865619 * r6865620;
        return r6865621;
}
↓
double f(double v) {
        double r6865622 = 1.0;
        double r6865623 = v;
        double r6865624 = r6865623 * r6865623;
        double r6865625 = r6865622 - r6865624;
        double r6865626 = 2.0;
        double r6865627 = sqrt(r6865626);
        double r6865628 = cbrt(r6865627);
        double r6865629 = r6865625 * r6865628;
        double r6865630 = r6865629 * r6865628;
        double r6865631 = 4.0;
        double r6865632 = r6865628 / r6865631;
        double r6865633 = r6865630 * r6865632;
        double r6865634 = 3.0;
        double r6865635 = r6865624 * r6865634;
        double r6865636 = r6865622 - r6865635;
        double r6865637 = r6865635 + r6865622;
        double r6865638 = r6865636 * r6865637;
        double r6865639 = sqrt(r6865638);
        double r6865640 = r6865633 * r6865639;
        double r6865641 = sqrt(r6865637);
        double r6865642 = r6865640 / r6865641;
        return r6865642;
}

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

    \[\left(\frac{\sqrt{2.0}}{4.0} \cdot \sqrt{1.0 - 3.0 \cdot \left(v \cdot v\right)}\right) \cdot \left(1.0 - v \cdot v\right)\]
  2. Using strategy rm

  3. Applied flip--0.0

    \[\leadsto \left(\frac{\sqrt{2.0}}{4.0} \cdot \sqrt{\color{blue}{\frac{1.0 \cdot 1.0 - \left(3.0 \cdot \left(v \cdot v\right)\right) \cdot \left(3.0 \cdot \left(v \cdot v\right)\right)}{1.0 + 3.0 \cdot \left(v \cdot v\right)}}}\right) \cdot \left(1.0 - v \cdot v\right)\]

  4. Applied sqrt-div0.0

    \[\leadsto \left(\frac{\sqrt{2.0}}{4.0} \cdot \color{blue}{\frac{\sqrt{1.0 \cdot 1.0 - \left(3.0 \cdot \left(v \cdot v\right)\right) \cdot \left(3.0 \cdot \left(v \cdot v\right)\right)}}{\sqrt{1.0 + 3.0 \cdot \left(v \cdot v\right)}}}\right) \cdot \left(1.0 - v \cdot v\right)\]

  5. Applied associate-*r/0.0

    \[\leadsto \color{blue}{\frac{\frac{\sqrt{2.0}}{4.0} \cdot \sqrt{1.0 \cdot 1.0 - \left(3.0 \cdot \left(v \cdot v\right)\right) \cdot \left(3.0 \cdot \left(v \cdot v\right)\right)}}{\sqrt{1.0 + 3.0 \cdot \left(v \cdot v\right)}}} \cdot \left(1.0 - v \cdot v\right)\]

  6. Applied associate-*l/0.0

    \[\leadsto \color{blue}{\frac{\left(\frac{\sqrt{2.0}}{4.0} \cdot \sqrt{1.0 \cdot 1.0 - \left(3.0 \cdot \left(v \cdot v\right)\right) \cdot \left(3.0 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1.0 - v \cdot v\right)}{\sqrt{1.0 + 3.0 \cdot \left(v \cdot v\right)}}}\]

  7. Simplified0.0

    \[\leadsto \frac{\color{blue}{\left(\left(1.0 - v \cdot v\right) \cdot \frac{\sqrt{2.0}}{4.0}\right) \cdot \sqrt{\left(1.0 + 3.0 \cdot \left(v \cdot v\right)\right) \cdot \left(1.0 - 3.0 \cdot \left(v \cdot v\right)\right)}}}{\sqrt{1.0 + 3.0 \cdot \left(v \cdot v\right)}}\]
  8. Using strategy rm

  9. Applied *-un-lft-identity0.0

    \[\leadsto \frac{\left(\left(1.0 - v \cdot v\right) \cdot \frac{\sqrt{2.0}}{\color{blue}{1 \cdot 4.0}}\right) \cdot \sqrt{\left(1.0 + 3.0 \cdot \left(v \cdot v\right)\right) \cdot \left(1.0 - 3.0 \cdot \left(v \cdot v\right)\right)}}{\sqrt{1.0 + 3.0 \cdot \left(v \cdot v\right)}}\]

  10. Applied add-cube-cbrt0.0

    \[\leadsto \frac{\left(\left(1.0 - v \cdot v\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{\sqrt{2.0}} \cdot \sqrt[3]{\sqrt{2.0}}\right) \cdot \sqrt[3]{\sqrt{2.0}}}}{1 \cdot 4.0}\right) \cdot \sqrt{\left(1.0 + 3.0 \cdot \left(v \cdot v\right)\right) \cdot \left(1.0 - 3.0 \cdot \left(v \cdot v\right)\right)}}{\sqrt{1.0 + 3.0 \cdot \left(v \cdot v\right)}}\]

  11. Applied times-frac0.0

    \[\leadsto \frac{\left(\left(1.0 - v \cdot v\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{\sqrt{2.0}} \cdot \sqrt[3]{\sqrt{2.0}}}{1} \cdot \frac{\sqrt[3]{\sqrt{2.0}}}{4.0}\right)}\right) \cdot \sqrt{\left(1.0 + 3.0 \cdot \left(v \cdot v\right)\right) \cdot \left(1.0 - 3.0 \cdot \left(v \cdot v\right)\right)}}{\sqrt{1.0 + 3.0 \cdot \left(v \cdot v\right)}}\]

  12. Applied associate-*r*0.0

    \[\leadsto \frac{\color{blue}{\left(\left(\left(1.0 - v \cdot v\right) \cdot \frac{\sqrt[3]{\sqrt{2.0}} \cdot \sqrt[3]{\sqrt{2.0}}}{1}\right) \cdot \frac{\sqrt[3]{\sqrt{2.0}}}{4.0}\right)} \cdot \sqrt{\left(1.0 + 3.0 \cdot \left(v \cdot v\right)\right) \cdot \left(1.0 - 3.0 \cdot \left(v \cdot v\right)\right)}}{\sqrt{1.0 + 3.0 \cdot \left(v \cdot v\right)}}\]

  13. Simplified0.0

    \[\leadsto \frac{\left(\color{blue}{\left(\left(\left(1.0 - v \cdot v\right) \cdot \sqrt[3]{\sqrt{2.0}}\right) \cdot \sqrt[3]{\sqrt{2.0}}\right)} \cdot \frac{\sqrt[3]{\sqrt{2.0}}}{4.0}\right) \cdot \sqrt{\left(1.0 + 3.0 \cdot \left(v \cdot v\right)\right) \cdot \left(1.0 - 3.0 \cdot \left(v \cdot v\right)\right)}}{\sqrt{1.0 + 3.0 \cdot \left(v \cdot v\right)}}\]

  14. Final simplification0.0

    \[\leadsto \frac{\left(\left(\left(\left(1.0 - v \cdot v\right) \cdot \sqrt[3]{\sqrt{2.0}}\right) \cdot \sqrt[3]{\sqrt{2.0}}\right) \cdot \frac{\sqrt[3]{\sqrt{2.0}}}{4.0}\right) \cdot \sqrt{\left(1.0 - \left(v \cdot v\right) \cdot 3.0\right) \cdot \left(\left(v \cdot v\right) \cdot 3.0 + 1.0\right)}}{\sqrt{\left(v \cdot v\right) \cdot 3.0 + 1.0}}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))