Average Error: 13.7 → 3.2
Time: 9.5m
Precision: 64
Internal Precision: 384
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{1}{t}}{\pi \cdot \sqrt{2}} - \frac{\frac{4}{t}}{\sqrt{2}} \cdot \left(\frac{{v}^{4}}{\pi} + \frac{v}{\frac{\pi}{v}}\right)\]

Error

Bits error versus v

Bits error versus t

Derivation

  1. Initial program 13.7

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

  2. Taylor expanded around 0 13.5

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

  3. Taylor expanded around 0 3.3

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

  4. Applied simplify3.2

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

Runtime

Time bar (total: 9.5m)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit +o reduce:binary-search
(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)))))