Average Error: 33.9 → 19.6
Time: 3.7m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 5.675529325076035 \cdot 10^{-249}:\\ \;\;\;\;\frac{\frac{\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{\sqrt[3]{-c} \cdot \sqrt[3]{-c}} \cdot \sqrt[3]{\frac{\sqrt[3]{-c}}{\sqrt{(\left(-c\right) \cdot \left(3 \cdot a\right) + \left(b \cdot b\right))_*} + b}}\right) \cdot \sqrt[3]{\frac{-c}{\sqrt{(\left(-c\right) \cdot \left(3 \cdot a\right) + \left(b \cdot b\right))_*} + b}}\right) \cdot \sqrt[3]{\frac{-c}{\sqrt{(\left(-c\right) \cdot \left(3 \cdot a\right) + \left(b \cdot b\right))_*} + b}}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Split input into 2 regimes

  2. if b < 5.675529325076035e-249

    1. Initial program 20.0

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

    2. Applied simplify20.0

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

    4. Applied associate-/r*20.0

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

    if 5.675529325076035e-249 < b

    1. Initial program 46.6

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

    2. Applied simplify46.6

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

    4. Applied flip--46.7

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

    5. Applied simplify24.2

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

    7. Applied add-cube-cbrt24.6

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

    8. Applied simplify24.6

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

    9. Applied simplify19.3

      \[\leadsto \left(\sqrt[3]{\frac{\frac{-c}{1}}{\sqrt{(\left(-c\right) \cdot \left(3 \cdot a\right) + \left(b \cdot b\right))_*} + b}} \cdot \sqrt[3]{\frac{\frac{-c}{1}}{\sqrt{(\left(-c\right) \cdot \left(3 \cdot a\right) + \left(b \cdot b\right))_*} + b}}\right) \cdot \color{blue}{\sqrt[3]{\frac{\left(-1\right) \cdot c}{\sqrt{(\left(-c\right) \cdot \left(3 \cdot a\right) + \left(b \cdot b\right))_*} + b}}}\]
    10. Using strategy rm

    11. Applied *-un-lft-identity19.3

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

    12. Applied add-cube-cbrt19.2

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

    13. Applied times-frac19.2

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

    14. Applied cbrt-prod19.3

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

    15. Applied simplify19.3

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

    16. Applied simplify19.3

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

  4. Applied simplify19.6

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \le 5.675529325076035 \cdot 10^{-249}:\\ \;\;\;\;\frac{\frac{\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{\sqrt[3]{-c} \cdot \sqrt[3]{-c}} \cdot \sqrt[3]{\frac{\sqrt[3]{-c}}{\sqrt{(\left(-c\right) \cdot \left(3 \cdot a\right) + \left(b \cdot b\right))_*} + b}}\right) \cdot \sqrt[3]{\frac{-c}{\sqrt{(\left(-c\right) \cdot \left(3 \cdot a\right) + \left(b \cdot b\right))_*} + b}}\right) \cdot \sqrt[3]{\frac{-c}{\sqrt{(\left(-c\right) \cdot \left(3 \cdot a\right) + \left(b \cdot b\right))_*} + b}}\\ \end{array}}\]

Runtime

Time bar (total: 3.7m)Debug logProfile

herbie shell --seed '#(1071215679 2002590028 935158157 1944352234 2656991306 2955288481)' +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))