Average Error: 32.8 → 8.4
Time: 1.5m
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.542974251663252 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{c}{\frac{b}{\frac{3}{2}}} - \frac{b + b}{a}}{3}\\ \mathbf{if}\;b \le 7.58777237353481 \cdot 10^{-166}:\\ \;\;\;\;\frac{1}{3} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{a}\\ \mathbf{if}\;b \le 2.492505129320434 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{3 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{\frac{-3}{2}}{3}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Split input into 4 regimes

  2. if b < -6.542974251663252e+143

    1. Initial program 56.5

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

    3. Applied *-un-lft-identity56.5

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

    4. Applied times-frac56.5

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

    5. Taylor expanded around -inf 11.6

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

    6. Applied simplify3.0

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

    if -6.542974251663252e+143 < b < 7.58777237353481e-166

    1. Initial program 9.5

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

    3. Applied *-un-lft-identity9.5

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

    4. Applied times-frac9.7

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

    if 7.58777237353481e-166 < b < 2.492505129320434e+69

    1. Initial program 36.0

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

    3. Applied flip-+36.1

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

    4. Applied simplify15.5

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

    if 2.492505129320434e+69 < b

    1. Initial program 57.3

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

    2. Taylor expanded around inf 14.6

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

    3. Applied simplify3.5

      \[\leadsto \color{blue}{\frac{c}{b} \cdot \frac{\frac{-3}{2}}{3}}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed '#(1064397287 3527694221 3797617954 1138343853 2854031332 1153838279)' 
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))