Average Error: 33.9 → 6.8
Time: 1.5m
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.008855815134272 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{b}{a}}{\frac{-3}{2}}\\ \mathbf{if}\;b \le 3.146955685671999 \cdot 10^{-302}:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}\\ \mathbf{if}\;b \le 3.437425097114325 \cdot 10^{+111}:\\ \;\;\;\;\frac{1}{3} \cdot \frac{3 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{\frac{3}{2} \cdot a}{\frac{b}{c}} + \left(\left(-b\right) - b\right)}\\ \end{array}\]

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 < -1.008855815134272e+152

    1. Initial program 60.1

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

    3. Applied flip-+62.3

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

    4. Applied simplify62.5

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

    6. Applied *-un-lft-identity62.5

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

    7. Applied times-frac62.5

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

    8. Applied simplify62.5

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

    9. Taylor expanded around -inf 24.9

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

    10. Applied simplify3.2

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

    if -1.008855815134272e+152 < b < 3.146955685671999e-302

    1. Initial program 8.8

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

    3. Applied associate-/r*8.8

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

    if 3.146955685671999e-302 < b < 3.437425097114325e+111

    1. Initial program 33.4

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

    3. Applied flip-+33.5

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

    4. Applied simplify16.9

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

    6. Applied *-un-lft-identity16.9

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

    7. Applied times-frac16.9

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

    8. Applied simplify9.3

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

    if 3.437425097114325e+111 < b

    1. Initial program 59.5

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

    3. Applied flip-+59.5

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

    4. Applied simplify33.2

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

    5. Taylor expanded around inf 14.7

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

    6. Applied simplify2.2

      \[\leadsto \color{blue}{\frac{c}{\frac{\frac{3}{2} \cdot a}{\frac{b}{c}} + \left(\left(-b\right) - b\right)}}\]
  3. Recombined 4 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 1.5m)Debug log

herbie shell --seed '#(2961832646 520228599 1275628947 1047906571 1774476463 2890033825)' 
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))