Average Error: 0.1 → 0.1
Time: 3.5m
Precision: 64
Internal Precision: 128
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) + rand \cdot \frac{\frac{a - \frac{1.0}{3.0}}{3}}{\sqrt{a - \frac{1.0}{3.0}}}\]

Error

Bits error versus a

Bits error versus rand

Derivation

  1. Initial program 0.1

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

  2. Simplified0.1

    \[\leadsto \color{blue}{rand \cdot \frac{a - \frac{1.0}{3.0}}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} + \left(a - \frac{1.0}{3.0}\right)}\]
  3. Using strategy rm

  4. Applied sqrt-prod0.1

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

  5. Applied associate-/r*0.1

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

  6. Final simplification0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) + rand \cdot \frac{\frac{a - \frac{1.0}{3.0}}{3}}{\sqrt{a - \frac{1.0}{3.0}}}\]

Reproduce

herbie shell --seed 2019094 
(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))))