Average Error: 0.0 → 0.0
Time: 9.6s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(\sqrt{1 - \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot 9} \cdot \frac{\sqrt{2}}{4}\right)}{\sqrt{\mathsf{fma}\left(v \cdot v, 3, 1\right)} + \sqrt{\mathsf{fma}\left(v \cdot v, 3, 1\right)} \cdot \left(v \cdot v\right)}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{\left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(\sqrt{1 - \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot 9} \cdot \frac{\sqrt{2}}{4}\right)}{\sqrt{\mathsf{fma}\left(v \cdot v, 3, 1\right)} + \sqrt{\mathsf{fma}\left(v \cdot v, 3, 1\right)} \cdot \left(v \cdot v\right)}
double f(double v) {
        double r6625689 = 2.0;
        double r6625690 = sqrt(r6625689);
        double r6625691 = 4.0;
        double r6625692 = r6625690 / r6625691;
        double r6625693 = 1.0;
        double r6625694 = 3.0;
        double r6625695 = v;
        double r6625696 = r6625695 * r6625695;
        double r6625697 = r6625694 * r6625696;
        double r6625698 = r6625693 - r6625697;
        double r6625699 = sqrt(r6625698);
        double r6625700 = r6625692 * r6625699;
        double r6625701 = r6625693 - r6625696;
        double r6625702 = r6625700 * r6625701;
        return r6625702;
}
↓
double f(double v) {
        double r6625703 = 1.0;
        double r6625704 = v;
        double r6625705 = r6625704 * r6625704;
        double r6625706 = r6625705 * r6625705;
        double r6625707 = r6625703 - r6625706;
        double r6625708 = 9.0;
        double r6625709 = r6625706 * r6625708;
        double r6625710 = r6625703 - r6625709;
        double r6625711 = sqrt(r6625710);
        double r6625712 = 2.0;
        double r6625713 = sqrt(r6625712);
        double r6625714 = 4.0;
        double r6625715 = r6625713 / r6625714;
        double r6625716 = r6625711 * r6625715;
        double r6625717 = r6625707 * r6625716;
        double r6625718 = 3.0;
        double r6625719 = fma(r6625705, r6625718, r6625703);
        double r6625720 = sqrt(r6625719);
        double r6625721 = r6625720 * r6625705;
        double r6625722 = r6625720 + r6625721;
        double r6625723 = r6625717 / r6625722;
        return r6625723;
}

Error

Bits error versus v

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 flip--0.0

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

  4. Applied flip--0.0

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

  5. Applied sqrt-div0.0

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

  6. Applied associate-*r/0.0

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

  7. Applied frac-times0.0

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

  8. Simplified0.0

    \[\leadsto \frac{\color{blue}{\left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 9 \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)}\right)}}{\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 + v \cdot v\right)}\]

  9. Simplified0.0

    \[\leadsto \frac{\left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 9 \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)}\right)}{\color{blue}{\sqrt{\mathsf{fma}\left(v \cdot v, 3, 1\right)} + \left(v \cdot v\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, 3, 1\right)}}}\]

  10. Final simplification0.0

    \[\leadsto \frac{\left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(\sqrt{1 - \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot 9} \cdot \frac{\sqrt{2}}{4}\right)}{\sqrt{\mathsf{fma}\left(v \cdot v, 3, 1\right)} + \sqrt{\mathsf{fma}\left(v \cdot v, 3, 1\right)} \cdot \left(v \cdot v\right)}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))