Average Error: 33.8 → 7.3
Time: 2.1m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.2909681847340677 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(c \cdot \frac{3}{2}\right) \cdot \frac{a}{b} - \left(b + b\right)}{a \cdot 3}\\ \mathbf{if}\;b \le 2.4554294536540332 \cdot 10^{-157}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 3}{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}}\\ \mathbf{if}\;b \le 5.089127204391984 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{c}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}} \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}}}}{3} \cdot \frac{3}{\left(\sqrt[3]{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}}} \cdot \sqrt[3]{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}}}\right) \cdot \sqrt[3]{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{\frac{-3}{2}}{3}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if b < -1.2909681847340677e+154

    1. Initial program 60.9

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

    2. Taylor expanded around -inf 11.6

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

    3. Applied simplify2.9

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

    if -1.2909681847340677e+154 < b < 2.4554294536540332e-157

    1. Initial program 10.4

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

    3. Applied clear-num10.5

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

    4. Applied simplify10.5

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

    if 2.4554294536540332e-157 < b < 5.089127204391984e+118

    1. Initial program 38.9

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

    3. Applied flip-+39.0

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

    4. Applied simplify16.4

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

    6. Applied add-cube-cbrt17.0

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

    7. Applied times-frac14.5

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

    8. Applied times-frac8.1

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

    9. Applied simplify7.5

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

    11. Applied add-cube-cbrt7.8

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

    if 5.089127204391984e+118 < b

    1. Initial program 59.8

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

    2. Taylor expanded around inf 15.1

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

    3. Applied simplify2.8

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

  4. Applied simplify7.3

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \le -1.2909681847340677 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(c \cdot \frac{3}{2}\right) \cdot \frac{a}{b} - \left(b + b\right)}{a \cdot 3}\\ \mathbf{if}\;b \le 2.4554294536540332 \cdot 10^{-157}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 3}{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}}\\ \mathbf{if}\;b \le 5.089127204391984 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{c}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}} \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}}}}{3} \cdot \frac{3}{\left(\sqrt[3]{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}}} \cdot \sqrt[3]{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}}}\right) \cdot \sqrt[3]{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{\frac{-3}{2}}{3}\\ \end{array}}\]

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed 2018201 
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))