Average Error: 33.6 → 6.8
Time: 2.7m
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.4483268212204236 \cdot 10^{+113}:\\ \;\;\;\;\frac{\frac{-2}{3} \cdot b}{a}\\ \mathbf{if}\;b \le 1.1160190105796002 \cdot 10^{-221}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 3} - b}{3}}{a}\\ \mathbf{if}\;b \le 2.881702391395132 \cdot 10^{+120}:\\ \;\;\;\;\left(\sqrt[3]{\frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}} \cdot \sqrt[3]{\frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}\right) \cdot \sqrt[3]{\frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{2}}{3} \cdot \frac{c}{b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Split input into 4 regimes

  2. if b < -1.4483268212204236e+113

    1. Initial program 47.8

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

    3. Applied associate-/r*47.8

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

    4. Applied simplify47.8

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

    5. Taylor expanded around -inf 3.9

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

    if -1.4483268212204236e+113 < b < 1.1160190105796002e-221

    1. Initial program 9.7

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

    3. Applied associate-/r*9.7

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

    4. Applied simplify9.7

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

    if 1.1160190105796002e-221 < b < 2.881702391395132e+120

    1. Initial program 36.6

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

    3. Applied flip-+36.7

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

    4. Applied simplify16.1

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

    6. Applied add-cube-cbrt16.8

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

    7. Applied simplify16.8

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

    8. Applied simplify8.6

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

    if 2.881702391395132e+120 < b

    1. Initial program 60.0

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

    2. Taylor expanded around inf 13.1

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

    3. Applied simplify1.8

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} \cdot \frac{\frac{-3}{2}}{3}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Applied simplify6.8

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \le -1.4483268212204236 \cdot 10^{+113}:\\ \;\;\;\;\frac{\frac{-2}{3} \cdot b}{a}\\ \mathbf{if}\;b \le 1.1160190105796002 \cdot 10^{-221}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 3} - b}{3}}{a}\\ \mathbf{if}\;b \le 2.881702391395132 \cdot 10^{+120}:\\ \;\;\;\;\left(\sqrt[3]{\frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}} \cdot \sqrt[3]{\frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}\right) \cdot \sqrt[3]{\frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{2}}{3} \cdot \frac{c}{b}\\ \end{array}}\]

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed '#(1070864556 424010669 783715395 1203517814 4070606583 4107618214)' 
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))