Average Error: 1.0 → 0.2
Time: 2.0m
Precision: 64
Internal Precision: 384
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\left(\left(\frac{{v}^{4}}{\sqrt{2} \cdot \frac{\pi}{\frac{4}{3}}} + \frac{\frac{{v}^{4}}{\frac{\pi}{18}}}{{\left(\sqrt{2}\right)}^{5}}\right) + \frac{\frac{4}{2}}{\sqrt{2}} \cdot \left(\frac{v}{\pi} \cdot v + \frac{{v}^{4}}{\pi}\right)\right) + \frac{\frac{4}{3}}{\pi} \cdot \left(\frac{1}{\sqrt{2}} + \frac{v \cdot v}{\sqrt{2}}\right)\]

Error

Bits error versus v

Derivation

  1. Initial program 1.0

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

  2. Taylor expanded around 0 0.2

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

  3. Applied simplify0.2

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

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1063027428 1192549564 1443466578 604016274 3637110559 1698629644)' 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))