Average Error: 33.7 → 20.0
Time: 2.9m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;-b \le -3.058392593590901 \cdot 10^{-134}:\\ \;\;\;\;\frac{\sqrt[3]{-c} \cdot \sqrt[3]{-c}}{\sqrt{b + \sqrt{(\left(-c\right) \cdot \left(3 \cdot a\right) + \left(b \cdot b\right))_*}}} \cdot \frac{\sqrt[3]{-c}}{\sqrt{\sqrt{(\left(-c\right) \cdot \left(3 \cdot a\right) + \left(b \cdot b\right))_*} + b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}{3 \cdot a}\\ \end{array}\]

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) < -3.058392593590901e-134

    1. Initial program 50.3

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

    2. Applied simplify50.3

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

    4. Applied *-un-lft-identity50.3

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

    5. Applied times-frac50.3

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

    7. Applied div-inv50.3

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

    8. Applied associate-*r*50.3

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

    10. Applied flip--50.3

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

    11. Applied associate-*r/50.3

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

    12. Applied associate-*l/50.3

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

    13. Applied simplify22.9

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

    15. Applied add-sqr-sqrt23.1

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

    16. Applied add-cube-cbrt23.5

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

    17. Applied times-frac23.5

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

    18. Applied simplify23.4

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

    19. Applied simplify19.0

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

    if -3.058392593590901e-134 < (- b)

    1. Initial program 20.8

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

    2. Applied simplify20.8

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

Runtime

Time bar (total: 2.9m)Debug logProfile

herbie shell --seed 2018199 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))