Average Error: 0.1 → 0.1
Time: 6.0m
Precision: 64
Internal Precision: 128
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[(\left(\frac{1}{\left|\sqrt[3]{9}\right|} \cdot \frac{rand}{\sqrt{\sqrt[3]{9} \cdot \left(a - \frac{1.0}{3.0}\right)}}\right) \cdot \left(a - \frac{1.0}{3.0}\right) + \left(a - \frac{1.0}{3.0}\right))_*\]

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 add-cube-cbrt0.1

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

  5. Applied associate-*l*0.1

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

  7. Applied sqrt-prod0.1

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

  8. Applied *-un-lft-identity0.1

    \[\leadsto (\left(\frac{\color{blue}{1 \cdot rand}}{\sqrt{\sqrt[3]{9} \cdot \sqrt[3]{9}} \cdot \sqrt{\sqrt[3]{9} \cdot \left(a - \frac{1.0}{3.0}\right)}}\right) \cdot \left(a - \frac{1.0}{3.0}\right) + \left(a - \frac{1.0}{3.0}\right))_*\]

  9. Applied times-frac0.1

    \[\leadsto (\color{blue}{\left(\frac{1}{\sqrt{\sqrt[3]{9} \cdot \sqrt[3]{9}}} \cdot \frac{rand}{\sqrt{\sqrt[3]{9} \cdot \left(a - \frac{1.0}{3.0}\right)}}\right)} \cdot \left(a - \frac{1.0}{3.0}\right) + \left(a - \frac{1.0}{3.0}\right))_*\]

  10. Simplified0.1

    \[\leadsto (\left(\color{blue}{\frac{1}{\left|\sqrt[3]{9}\right|}} \cdot \frac{rand}{\sqrt{\sqrt[3]{9} \cdot \left(a - \frac{1.0}{3.0}\right)}}\right) \cdot \left(a - \frac{1.0}{3.0}\right) + \left(a - \frac{1.0}{3.0}\right))_*\]

  11. Final simplification0.1

    \[\leadsto (\left(\frac{1}{\left|\sqrt[3]{9}\right|} \cdot \frac{rand}{\sqrt{\sqrt[3]{9} \cdot \left(a - \frac{1.0}{3.0}\right)}}\right) \cdot \left(a - \frac{1.0}{3.0}\right) + \left(a - \frac{1.0}{3.0}\right))_*\]

Reproduce

herbie shell --seed 2019090 +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))))