\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

↓

\[\frac{\sqrt{(\left(a \cdot -3\right) \cdot c + \left(b \cdot b\right))_*} - b}{3 \cdot a}\]

Error

The X axis uses an exponential scale — Bits error versus `a`

Derivation

Initial program 28.4
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
Initial simplification28.4
\[\leadsto \frac{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}{3 \cdot a}\]
Taylor expanded around -inf 28.4
\[\leadsto \frac{\sqrt{\color{blue}{{b}^{2} - 3 \cdot \left(a \cdot c\right)}} - b}{3 \cdot a}\]
Simplified28.4
\[\leadsto \frac{\sqrt{\color{blue}{(\left(a \cdot -3\right) \cdot c + \left(b \cdot b\right))_*}} - b}{3 \cdot a}\]
Final simplification28.4
\[\leadsto \frac{\sqrt{(\left(a \cdot -3\right) \cdot c + \left(b \cdot b\right))_*} - b}{3 \cdot a}\]

Runtime

herbie shell --seed 2018251 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))