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

↓

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

Error

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

Derivation

Initial program 28.8
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
Applied simplify28.8
\[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} - b}{3 \cdot a}}\]
Using strategy rm
Applied add-cube-cbrt28.8
\[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{\left(\sqrt[3]{\left(a \cdot c\right) \cdot 3} \cdot \sqrt[3]{\left(a \cdot c\right) \cdot 3}\right) \cdot \sqrt[3]{\left(a \cdot c\right) \cdot 3}}} - b}{3 \cdot a}\]

Runtime

herbie shell --seed '#(1071215679 2002590028 935158157 1944352234 2656991306 2955288481)' 
(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)))