Average Error: 0.0 → 0.0
Time: 18.4s
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 r6778266 = 2.0;
        double r6778267 = sqrt(r6778266);
        double r6778268 = 4.0;
        double r6778269 = r6778267 / r6778268;
        double r6778270 = 1.0;
        double r6778271 = 3.0;
        double r6778272 = v;
        double r6778273 = r6778272 * r6778272;
        double r6778274 = r6778271 * r6778273;
        double r6778275 = r6778270 - r6778274;
        double r6778276 = sqrt(r6778275);
        double r6778277 = r6778269 * r6778276;
        double r6778278 = r6778270 - r6778273;
        double r6778279 = r6778277 * r6778278;
        return r6778279;
}
↓
double f(double v) {
        double r6778280 = 2.0;
        double r6778281 = sqrt(r6778280);
        double r6778282 = -3.0;
        double r6778283 = v;
        double r6778284 = r6778283 * r6778283;
        double r6778285 = 1.0;
        double r6778286 = fma(r6778282, r6778284, r6778285);
        double r6778287 = sqrt(r6778286);
        double r6778288 = r6778281 * r6778287;
        double r6778289 = r6778284 * r6778284;
        double r6778290 = r6778285 - r6778289;
        double r6778291 = r6778288 * r6778290;
        double r6778292 = r6778291 * r6778291;
        double r6778293 = r6778291 * r6778292;
        double r6778294 = 4.0;
        double r6778295 = fma(r6778284, r6778294, r6778294);
        double r6778296 = r6778295 * r6778295;
        double r6778297 = r6778295 * r6778296;
        double r6778298 = r6778293 / r6778297;
        double r6778299 = cbrt(r6778298);
        return r6778299;
}

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))))