Derivation

Split input into 2 regimes
if b < -8.11375828407908e-162
1. Initial program 22.6
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
if -8.11375828407908e-162 < b
1. Initial program 38.6
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
2. Applied simplify38.6
  \[\leadsto \color{blue}{\frac{\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}{3 \cdot a}}\]
3. Using strategy rm
4. Applied flip--38.8
  \[\leadsto \frac{\color{blue}{\frac{\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} \cdot \sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b \cdot b}{\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}}}{3 \cdot a}\]
5. Applied simplify21.2
  \[\leadsto \frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot \left(-3\right)}}{\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}}{3 \cdot a}\]
6. Taylor expanded around 0 21.1
  \[\leadsto \frac{\frac{\left(c \cdot a\right) \cdot \left(-3\right)}{\sqrt{\color{blue}{{b}^{2} - 3 \cdot \left(c \cdot a\right)}} + b}}{3 \cdot a}\]
7. Applied simplify16.8
  \[\leadsto \color{blue}{\frac{\frac{-c}{1}}{\sqrt{(\left(-c\right) \cdot \left(3 \cdot a\right) + \left(b \cdot b\right))_*} + b}}\]
8. Taylor expanded around 0 16.8
  \[\leadsto \frac{\frac{-c}{1}}{\sqrt{\color{blue}{{b}^{2} - 3 \cdot \left(c \cdot a\right)}} + b}\]
9. Applied simplify16.8
  \[\leadsto \color{blue}{\frac{-c}{b + \sqrt{(\left(-3\right) \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}}\]
Recombined 2 regimes into one program.

Runtime

herbie shell --seed 2018166 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))

Error

Derivation

`if b < -8.11375828407908e-162`

`if -8.11375828407908e-162 < b`

Runtime