Average Error: 0.4 → 0.3
Time: 2.0m
Precision: 64
Internal Precision: 384
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[(\left(\frac{3}{\left(t + t\right) \cdot \sqrt{2}}\right) \cdot \left(\frac{v}{\frac{\pi}{v}}\right) + \left((\left(\frac{\frac{27}{2}}{\pi}\right) \cdot \left(\frac{\frac{{v}^{4}}{t}}{{\left(\sqrt{2}\right)}^{5}}\right) + \left(\frac{\frac{\frac{1}{\pi}}{\sqrt{2}}}{t}\right))_*\right))_* - (\left(\frac{12}{\left(t + t\right) \cdot \sqrt{2}}\right) \cdot \left(\frac{{v}^{4}}{\pi}\right) + \left((\left(\frac{v}{\pi}\right) \cdot v + \left(\frac{{v}^{4}}{\pi}\right))_* \cdot \frac{\frac{4}{t}}{\sqrt{2}}\right))_*\]

Error

Bits error versus v

Bits error versus t

Derivation

  1. Initial program 0.4

    \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]

  2. Taylor expanded around 0 0.6

    \[\leadsto \color{blue}{\left(\frac{27}{2} \cdot \frac{{v}^{4}}{\pi \cdot \left(t \cdot {\left(\sqrt{2}\right)}^{5}\right)} + \left(3 \cdot \frac{{v}^{2}}{\pi \cdot \left(t \cdot {\left(\sqrt{2}\right)}^{3}\right)} + \frac{1}{\pi \cdot \left(t \cdot \sqrt{2}\right)}\right)\right) - \left(12 \cdot \frac{{v}^{4}}{\pi \cdot \left(t \cdot {\left(\sqrt{2}\right)}^{3}\right)} + \left(4 \cdot \frac{{v}^{4}}{\pi \cdot \left(t \cdot \sqrt{2}\right)} + 4 \cdot \frac{{v}^{2}}{\pi \cdot \left(t \cdot \sqrt{2}\right)}\right)\right)}\]

  3. Applied simplify0.5

    \[\leadsto \color{blue}{(\left(\frac{\frac{3}{2}}{t \cdot \sqrt{2}}\right) \cdot \left(\frac{v \cdot v}{\pi}\right) + \left((\left(\frac{\frac{27}{2}}{\pi}\right) \cdot \left(\frac{\frac{{v}^{4}}{t}}{{\left(\sqrt{2}\right)}^{5}}\right) + \left(\frac{\frac{1}{\pi}}{t \cdot \sqrt{2}}\right))_*\right))_* - (\left(\frac{12}{\sqrt{2} \cdot \left(t + t\right)}\right) \cdot \left(\frac{{v}^{4}}{\pi}\right) + \left(\frac{\frac{4}{t}}{\pi} \cdot \left(\frac{v \cdot v}{\sqrt{2}} + \frac{{v}^{4}}{\sqrt{2}}\right)\right))_*}\]

  4. Taylor expanded around 0 0.5

    \[\leadsto (\left(\frac{\frac{3}{2}}{t \cdot \sqrt{2}}\right) \cdot \left(\frac{v \cdot v}{\pi}\right) + \left((\left(\frac{\frac{27}{2}}{\pi}\right) \cdot \left(\frac{\frac{{v}^{4}}{t}}{{\left(\sqrt{2}\right)}^{5}}\right) + \left(\frac{\frac{1}{\pi}}{t \cdot \sqrt{2}}\right))_*\right))_* - (\left(\frac{12}{\sqrt{2} \cdot \left(t + t\right)}\right) \cdot \left(\frac{{v}^{4}}{\pi}\right) + \color{blue}{\left(4 \cdot \frac{{v}^{4}}{\pi \cdot \left(t \cdot \sqrt{2}\right)} + 4 \cdot \frac{{v}^{2}}{\pi \cdot \left(t \cdot \sqrt{2}\right)}\right)})_*\]

  5. Applied simplify0.3

    \[\leadsto \color{blue}{(\left(\frac{3}{\left(t + t\right) \cdot \sqrt{2}}\right) \cdot \left(\frac{v}{\frac{\pi}{v}}\right) + \left((\left(\frac{\frac{27}{2}}{\pi}\right) \cdot \left(\frac{\frac{{v}^{4}}{t}}{{\left(\sqrt{2}\right)}^{5}}\right) + \left(\frac{\frac{\frac{1}{\pi}}{\sqrt{2}}}{t}\right))_*\right))_* - (\left(\frac{12}{\left(t + t\right) \cdot \sqrt{2}}\right) \cdot \left(\frac{{v}^{4}}{\pi}\right) + \left((\left(\frac{v}{\pi}\right) \cdot v + \left(\frac{{v}^{4}}{\pi}\right))_* \cdot \frac{\frac{4}{t}}{\sqrt{2}}\right))_*}\]

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +o rules:numerics
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))