Error

Derivation

1. Initial program 0.5

   \[\left(a - \left(\frac{\left(real->posit(1.0)\right)}{\left(real->posit(3.0)\right)}\right)\right) \cdot \left(\frac{\left(real->posit(1)\right)}{\left(\left(\frac{\left(real->posit(1)\right)}{\left(\sqrt{\left(\left(real->posit(9)\right) \cdot \left(a - \left(\frac{\left(real->posit(1.0)\right)}{\left(real->posit(3.0)\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]
2. Using strategy rm

3. Applied distribute-rgt-in0.5

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

5. Applied sub-neg0.5

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

6. Applied distribute-lft-in0.5

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

7. Applied associate-+l+0.4

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

9. Applied associate-*l/0.4

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

10. Applied associate-*l/0.5

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

12. Applied associate-/l*0.4

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

13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019051 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (*.p16 (-.p16 a (/.p16 (real->posit16 1.0) (real->posit16 3.0))) (+.p16 (real->posit16 1) (*.p16 (/.p16 (real->posit16 1) (sqrt.p16 (*.p16 (real->posit16 9) (-.p16 a (/.p16 (real->posit16 1.0) (real->posit16 3.0)))))) rand))))