Average Error: 1.0 → 0.0

Time: 32.5s

Precision: 64

Internal Precision: 128

\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

↓

\[\frac{\frac{\frac{4}{3}}{\pi - \left(v \cdot v\right) \cdot \pi}}{\sqrt{2 + v \cdot \left(v \cdot -6\right)}}\]

Error

The X axis uses an exponential scale — Bits error versus `v`

Derivation

Initial program 1.0
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
Simplified0.0
\[\leadsto \color{blue}{\frac{\frac{\frac{4}{3}}{\pi - \left(v \cdot v\right) \cdot \pi}}{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}}\]
Final simplification0.0
\[\leadsto \frac{\frac{\frac{4}{3}}{\pi - \left(v \cdot v\right) \cdot \pi}}{\sqrt{2 + v \cdot \left(v \cdot -6\right)}}\]

Reproduce

herbie shell --seed 2019068 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))