Average Error: 0.1 → 0.1

Time: 1.4m

Precision: 64

\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]

↓

\[\frac{a - \frac{1.0}{3.0}}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}} \cdot rand + \left(a - \frac{1.0}{3.0}\right)\]

Error

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

Derivation

Initial program 0.1
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
Simplified0.1
\[\leadsto \color{blue}{rand \cdot \frac{a - \frac{1.0}{3.0}}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} + \left(a - \frac{1.0}{3.0}\right)}\]
Final simplification0.1
\[\leadsto \frac{a - \frac{1.0}{3.0}}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}} \cdot rand + \left(a - \frac{1.0}{3.0}\right)\]

Reproduce

herbie shell --seed 2019100 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand))))