Average Error: 12.7 → 0.3
Time: 17.5s
Precision: 64
Internal precision: 384
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - {v}^2\right)\right) \cdot \sqrt{2 - 6 \cdot {v}^2}}\]
⬇
\[\left(\frac{\frac{\frac{4}{3}}{\pi}}{\sqrt{2}} + \frac{\frac{v}{\sqrt{2}} \cdot \left(\frac{4}{2} + \frac{4}{3}\right)}{\frac{\pi}{v}}\right) + \left(\frac{{v}^{4} \cdot \frac{18}{\pi}}{{\left(\sqrt{2}\right)}^{5}} + \frac{{v}^{4} \cdot \left(\frac{4}{3} + \frac{4}{2}\right)}{\sqrt{2} \cdot \pi}\right)\]

Error

Bits error versus v

Derivation

  1. Initial program 12.7

    \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - {v}^2\right)\right) \cdot \sqrt{2 - 6 \cdot {v}^2}}\]

  2. Applied taylor 0.3

    \[\leadsto \frac{4}{3} \cdot \frac{{v}^2}{\pi \cdot \sqrt{2}} + \left(\frac{4}{3} \cdot \frac{1}{\pi \cdot \sqrt{2}} + \left(4 \cdot \frac{{v}^2}{\pi \cdot {\left(\sqrt{2}\right)}^{3}} + \left(4 \cdot \frac{{v}^{4}}{\pi \cdot {\left(\sqrt{2}\right)}^{3}} + \left(\frac{4}{3} \cdot \frac{{v}^{4}}{\pi \cdot \sqrt{2}} + 18 \cdot \frac{{v}^{4}}{\pi \cdot {\left(\sqrt{2}\right)}^{5}}\right)\right)\right)\right)\]

  3. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{\frac{4}{3} \cdot \frac{{v}^2}{\pi \cdot \sqrt{2}} + \left(\frac{4}{3} \cdot \frac{1}{\pi \cdot \sqrt{2}} + \left(4 \cdot \frac{{v}^2}{\pi \cdot {\left(\sqrt{2}\right)}^{3}} + \left(4 \cdot \frac{{v}^{4}}{\pi \cdot {\left(\sqrt{2}\right)}^{3}} + \left(\frac{4}{3} \cdot \frac{{v}^{4}}{\pi \cdot \sqrt{2}} + 18 \cdot \frac{{v}^{4}}{\pi \cdot {\left(\sqrt{2}\right)}^{5}}\right)\right)\right)\right)}\]

  4. Applied simplify 0.3

    \[\leadsto \color{blue}{\left(\frac{\frac{\frac{4}{3}}{\pi}}{\sqrt{2}} + \left(v \cdot \frac{v}{\pi}\right) \cdot \left(\frac{\frac{4}{3}}{\sqrt{2}} + \frac{\frac{4}{2}}{\sqrt{2}}\right)\right) + \left(\frac{\frac{{v}^{4}}{\frac{\pi}{18}}}{{\left(\sqrt{2}\right)}^{5}} + \frac{{v}^{4}}{\pi} \cdot \left(\frac{\frac{4}{2}}{\sqrt{2}} + \frac{\frac{4}{3}}{\sqrt{2}}\right)\right)}\]

  5. Applied simplify 0.3

    \[\leadsto \color{blue}{\left(\frac{\frac{\frac{4}{3}}{\pi}}{\sqrt{2}} + \frac{\frac{v}{\sqrt{2}} \cdot \left(\frac{4}{2} + \frac{4}{3}\right)}{\frac{\pi}{v}}\right)} + \left(\frac{\frac{{v}^{4}}{\frac{\pi}{18}}}{{\left(\sqrt{2}\right)}^{5}} + \frac{{v}^{4}}{\pi} \cdot \left(\frac{\frac{4}{2}}{\sqrt{2}} + \frac{\frac{4}{3}}{\sqrt{2}}\right)\right)\]

  6. Applied simplify 0.3

    \[\leadsto \left(\frac{\frac{\frac{4}{3}}{\pi}}{\sqrt{2}} + \frac{\frac{v}{\sqrt{2}} \cdot \left(\frac{4}{2} + \frac{4}{3}\right)}{\frac{\pi}{v}}\right) + \color{blue}{\left(\frac{{v}^{4} \cdot \frac{18}{\pi}}{{\left(\sqrt{2}\right)}^{5}} + \frac{{v}^{4} \cdot \left(\frac{4}{3} + \frac{4}{2}\right)}{\sqrt{2} \cdot \pi}\right)}\]
  7. Removed slow pow expressions

Runtime

Time bar (total: 17.5s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1064524629 4159152179 2999149171 575749698 4006532819 692958815)'
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (sqr v))) (sqrt (- 2 (* 6 (sqr v)))))))