Average Error: 0.0 → 0.0
Time: 19.9s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\sqrt[3]{\frac{\left(\left(\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right)}\right) \cdot \left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)\right) \cdot \left(\left(\left(\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right)}\right) \cdot \left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)\right) \cdot \left(\left(\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right)}\right) \cdot \left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)\right)\right)}{\mathsf{fma}\left(v \cdot v, 4, 4\right) \cdot \left(\mathsf{fma}\left(v \cdot v, 4, 4\right) \cdot \mathsf{fma}\left(v \cdot v, 4, 4\right)\right)}}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\sqrt[3]{\frac{\left(\left(\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right)}\right) \cdot \left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)\right) \cdot \left(\left(\left(\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right)}\right) \cdot \left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)\right) \cdot \left(\left(\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right)}\right) \cdot \left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)\right)\right)}{\mathsf{fma}\left(v \cdot v, 4, 4\right) \cdot \left(\mathsf{fma}\left(v \cdot v, 4, 4\right) \cdot \mathsf{fma}\left(v \cdot v, 4, 4\right)\right)}}
double f(double v) {
        double r7021710 = 2.0;
        double r7021711 = sqrt(r7021710);
        double r7021712 = 4.0;
        double r7021713 = r7021711 / r7021712;
        double r7021714 = 1.0;
        double r7021715 = 3.0;
        double r7021716 = v;
        double r7021717 = r7021716 * r7021716;
        double r7021718 = r7021715 * r7021717;
        double r7021719 = r7021714 - r7021718;
        double r7021720 = sqrt(r7021719);
        double r7021721 = r7021713 * r7021720;
        double r7021722 = r7021714 - r7021717;
        double r7021723 = r7021721 * r7021722;
        return r7021723;
}
↓
double f(double v) {
        double r7021724 = 2.0;
        double r7021725 = sqrt(r7021724);
        double r7021726 = -3.0;
        double r7021727 = v;
        double r7021728 = r7021727 * r7021727;
        double r7021729 = 1.0;
        double r7021730 = fma(r7021726, r7021728, r7021729);
        double r7021731 = sqrt(r7021730);
        double r7021732 = r7021725 * r7021731;
        double r7021733 = r7021728 * r7021728;
        double r7021734 = r7021729 - r7021733;
        double r7021735 = r7021732 * r7021734;
        double r7021736 = r7021735 * r7021735;
        double r7021737 = r7021735 * r7021736;
        double r7021738 = 4.0;
        double r7021739 = fma(r7021728, r7021738, r7021738);
        double r7021740 = r7021739 * r7021739;
        double r7021741 = r7021739 * r7021740;
        double r7021742 = r7021737 / r7021741;
        double r7021743 = cbrt(r7021742);
        return r7021743;
}

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 associate-*l/0.0

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

  5. Applied frac-times0.0

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

  6. Simplified0.0

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

  7. Simplified0.0

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

  9. Applied add-cbrt-cube0.0

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

  10. Applied add-cbrt-cube1.0

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

  11. Applied cbrt-undiv0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))