Average Error: 2.2 → 2.2
Time: 27.3s
Precision: 64
Internal Precision: 576
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2}}{a}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 2.2

    \[\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(real->posit(4)\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(real->posit(2)\right) \cdot a\right)}\]

  2. Simplified2.2

    \[\leadsto \color{blue}{\frac{\left(\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(c \cdot a\right) \cdot \left(real->posit(4)\right)\right)\right)}\right) - b\right)}{\left(\left(real->posit(2)\right) \cdot a\right)}}\]
  3. Using strategy rm

  4. Applied associate-/r*2.2

    \[\leadsto \color{blue}{\frac{\left(\frac{\left(\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(c \cdot a\right) \cdot \left(real->posit(4)\right)\right)\right)}\right) - b\right)}{\left(real->posit(2)\right)}\right)}{a}}\]

  5. Final simplification2.2

    \[\leadsto \frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2}}{a}\]

Reproduce

herbie shell --seed 2019089 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  (/.p16 (+.p16 (neg.p16 b) (sqrt.p16 (-.p16 (*.p16 b b) (*.p16 (real->posit16 4) (*.p16 a c))))) (*.p16 (real->posit16 2) a)))