Average Error: 33.6 → 10.3
Time: 24.2s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.9985891699139 \cdot 10^{+147}:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{(\frac{3}{2} \cdot \left(\frac{a}{\frac{b}{c}}\right) + \left(-2 \cdot b\right))_*}}\\ \mathbf{elif}\;b \le 3.357493650498016 \cdot 10^{-130}:\\ \;\;\;\;\frac{1}{3 \cdot a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\ \end{array}\]

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 < -4.9985891699139e+147

    1. Initial program 58.3

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

    3. Applied clear-num58.3

      \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}\]

    4. Taylor expanded around -inf 10.8

      \[\leadsto \frac{1}{\frac{3 \cdot a}{\color{blue}{\frac{3}{2} \cdot \frac{a \cdot c}{b} - 2 \cdot b}}}\]

    5. Simplified2.8

      \[\leadsto \frac{1}{\frac{3 \cdot a}{\color{blue}{(\frac{3}{2} \cdot \left(\frac{a}{\frac{b}{c}}\right) + \left(-2 \cdot b\right))_*}}}\]

    if -4.9985891699139e+147 < b < 3.357493650498016e-130

    1. Initial program 11.1

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

    3. Applied div-inv11.1

      \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{1}{3 \cdot a}}\]

    if 3.357493650498016e-130 < b

    1. Initial program 50.6

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

    2. Taylor expanded around inf 11.4

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.9985891699139 \cdot 10^{+147}:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{(\frac{3}{2} \cdot \left(\frac{a}{\frac{b}{c}}\right) + \left(-2 \cdot b\right))_*}}\\ \mathbf{elif}\;b \le 3.357493650498016 \cdot 10^{-130}:\\ \;\;\;\;\frac{1}{3 \cdot a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019093 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))