Average Error: 33.9 → 10.0
Time: 5.2m
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.7343202869522576 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{(\frac{3}{2} \cdot \left(\frac{a}{\frac{b}{c}}\right) + \left(b \cdot -2\right))_*}{3}}{a}\\ \mathbf{elif}\;b \le 1.2094793871458965 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} + \left(-b\right)}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Split input into 3 regimes

  2. if b < -1.7343202869522576e+143

    1. Initial program 56.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
    2. Using strategy rm

    3. Applied associate-/r*56.9

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

    4. Taylor expanded around -inf 10.8

      \[\leadsto \frac{\frac{\color{blue}{\frac{3}{2} \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{3}}{a}\]

    5. Simplified2.9

      \[\leadsto \frac{\frac{\color{blue}{(\frac{3}{2} \cdot \left(\frac{a}{\frac{b}{c}}\right) + \left(-2 \cdot b\right))_*}}{3}}{a}\]

    if -1.7343202869522576e+143 < b < 1.2094793871458965e-80

    1. Initial program 12.6

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
    2. Using strategy rm

    3. Applied associate-/r*12.6

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

    if 1.2094793871458965e-80 < b

    1. Initial program 52.8

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

    2. Taylor expanded around inf 9.0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.7343202869522576 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{(\frac{3}{2} \cdot \left(\frac{a}{\frac{b}{c}}\right) + \left(b \cdot -2\right))_*}{3}}{a}\\ \mathbf{elif}\;b \le 1.2094793871458965 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} + \left(-b\right)}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019088 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))