Average Error: 0.5 → 0.7
Time: 47.7s
Precision: 64
Internal Precision: 128
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{1}{\pi \cdot t}}{\sqrt{2}} - (\frac{5}{2} \cdot \left(\frac{v \cdot v}{\left(\pi \cdot t\right) \cdot \sqrt{2}}\right) + \left(\frac{\frac{53}{8}}{\left(\pi \cdot t\right) \cdot \sqrt{2}} \cdot {v}^{4}\right))_*\]

Error

Bits error versus v

Bits error versus t

Derivation

  1. Initial program 0.5

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019089 +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)))))