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

↓

\[\left(\frac{\frac{1}{\sqrt{2} \cdot \pi}}{t} - \frac{{v}^{4}}{\left(\sqrt{2} \cdot \pi\right) \cdot t} \cdot \frac{53}{8}\right) + \frac{-5}{2} \cdot \frac{v \cdot v}{\left(\sqrt{2} \cdot \pi\right) \cdot t}\]

Derivation

Initial program 0.4
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
Taylor expanded around 0 0.6
\[\leadsto \color{blue}{\frac{1}{t \cdot \left(\sqrt{2} \cdot \pi\right)} - \left(\frac{53}{8} \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + \frac{5}{2} \cdot \frac{{v}^{2}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)}\]
Simplified0.6
\[\leadsto \color{blue}{\frac{v \cdot v}{\left(\sqrt{2} \cdot \pi\right) \cdot t} \cdot \frac{-5}{2} + \left(\frac{1}{\left(\sqrt{2} \cdot \pi\right) \cdot t} - \frac{{v}^{4}}{\left(\sqrt{2} \cdot \pi\right) \cdot t} \cdot \frac{53}{8}\right)}\]
Using strategy rm
Applied associate-/r*0.4
\[\leadsto \frac{v \cdot v}{\left(\sqrt{2} \cdot \pi\right) \cdot t} \cdot \frac{-5}{2} + \left(\color{blue}{\frac{\frac{1}{\sqrt{2} \cdot \pi}}{t}} - \frac{{v}^{4}}{\left(\sqrt{2} \cdot \pi\right) \cdot t} \cdot \frac{53}{8}\right)\]
Final simplification0.4
\[\leadsto \left(\frac{\frac{1}{\sqrt{2} \cdot \pi}}{t} - \frac{{v}^{4}}{\left(\sqrt{2} \cdot \pi\right) \cdot t} \cdot \frac{53}{8}\right) + \frac{-5}{2} \cdot \frac{v \cdot v}{\left(\sqrt{2} \cdot \pi\right) \cdot t}\]

Reproduce

herbie shell --seed 2019050 
(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)))))

Error

Derivation

Reproduce