Average Error: 1.0 → 0.0
Time: 57.5s
Precision: 64
Internal Precision: 384
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{4}{3}}{\pi - \left(\pi \cdot v\right) \cdot v}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

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 1.0

    \[\leadsto \frac{4}{\color{blue}{\left(3 \cdot \pi - 3 \cdot \left(\pi \cdot {v}^{2}\right)\right)} \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

  3. Applied simplify0.0

    \[\leadsto \color{blue}{\frac{\frac{\frac{4}{3}}{\pi - \left(\pi \cdot v\right) \cdot v}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}}\]
  4. Removed slow pow expressions.

Runtime

Time bar (total: 57.5s)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit +o reduce:binary-search
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))