Average Error: 33.1 → 15.3
Time: 1.9m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 2.0019491190324444 \cdot 10^{-234}:\\ \;\;\;\;\frac{1}{3 \cdot \frac{a}{\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}}\\ \mathbf{elif}\;b \le 1.8559713529907816 \cdot 10^{+98}:\\ \;\;\;\;\frac{\frac{\left(3 \cdot a\right) \cdot \left(-c\right)}{3 \cdot \left(b + \sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b}{c} \cdot -2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Split input into 3 regimes

  2. if b < 2.0019491190324444e-234

    1. Initial program 20.4

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

    2. Initial simplification20.4

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

    4. Applied *-un-lft-identity20.4

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

    5. Applied associate-/l*20.4

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

    7. Applied *-un-lft-identity20.4

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

    8. Applied times-frac20.4

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

    9. Simplified20.4

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

    if 2.0019491190324444e-234 < b < 1.8559713529907816e+98

    1. Initial program 34.7

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

    2. Initial simplification34.7

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

    4. Applied associate-/r*34.7

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

    6. Applied flip--34.8

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

    7. Applied associate-/l/34.9

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

    8. Simplified16.4

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

    if 1.8559713529907816e+98 < b

    1. Initial program 58.7

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

    2. Initial simplification58.7

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

    4. Applied *-un-lft-identity58.7

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

    5. Applied associate-/l*58.7

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

    6. Taylor expanded around 0 3.2

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

  4. Final simplification15.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2.0019491190324444 \cdot 10^{-234}:\\ \;\;\;\;\frac{1}{3 \cdot \frac{a}{\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}}\\ \mathbf{elif}\;b \le 1.8559713529907816 \cdot 10^{+98}:\\ \;\;\;\;\frac{\frac{\left(3 \cdot a\right) \cdot \left(-c\right)}{3 \cdot \left(b + \sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b}{c} \cdot -2}\\ \end{array}\]

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2018230 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))