Average Error: 0.1 → 0.1
Time: 50.3s
Precision: 64
Internal Precision: 128
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\frac{1}{\frac{\sqrt{a - \frac{1.0}{3.0}}}{\frac{rand}{3}}} \cdot \left(a - \frac{1.0}{3.0}\right) + \left(a - \frac{1.0}{3.0}\right)\]

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

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

  4. Applied fma-udef0.1

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

  6. Applied sqrt-prod0.1

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

  7. Applied associate-/r*0.1

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

  9. Applied clear-num0.1

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

  10. Final simplification0.1

    \[\leadsto \frac{1}{\frac{\sqrt{a - \frac{1.0}{3.0}}}{\frac{rand}{3}}} \cdot \left(a - \frac{1.0}{3.0}\right) + \left(a - \frac{1.0}{3.0}\right)\]

Reproduce

herbie shell --seed 2019053 +o rules:numerics
(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))))