Average Error: 33.6 → 9.8
Time: 5.1s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.5592132314298372 \cdot 10^{118}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 2.82035357691290446 \cdot 10^{-33}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -2.5592132314298372e118

    1. Initial program 51.3

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

    2. Taylor expanded around -inf 3.3

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}}\]

    if -2.5592132314298372e118 < b < 2.82035357691290446e-33

    1. Initial program 13.4

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

    if 2.82035357691290446e-33 < b

    1. Initial program 54.7

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

    2. Taylor expanded around inf 7.4

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.5592132314298372 \cdot 10^{118}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 2.82035357691290446 \cdot 10^{-33}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020155 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (neg b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))