Average Error: 0.1 → 0.1
Time: 3.7m
Precision: 64
Internal Precision: 128
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) + rand \cdot \frac{\sqrt{a - \frac{1.0}{3.0}}}{3}\]

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 div-inv0.1

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

  7. Applied associate-*l*0.1

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

  9. Applied associate-*l/0.1

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

  10. Simplified0.1

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

  12. Applied sqrt-prod0.1

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

  13. Applied add-sqr-sqrt0.1

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

  14. Applied times-frac0.1

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

  15. Simplified0.1

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

  16. Simplified0.1

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

  17. Final simplification0.1

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

Reproduce

herbie shell --seed 2019094 +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))))