Average Error: 33.7 → 9.4
Time: 2.6m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{-3}{2}}{b}}{3} \le -1.048690060768616 \cdot 10^{-86}:\\ \;\;\;\;\frac{-1}{3} \cdot \frac{\frac{a \cdot \left(3 \cdot c\right)}{b + \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 3}}}{a}\\ \mathbf{if}\;\frac{\frac{\frac{-3}{2}}{b}}{3} \le 6.096801289818052 \cdot 10^{-306}:\\ \;\;\;\;\frac{\frac{-3}{2}}{3} \cdot \frac{c}{b}\\ \mathbf{if}\;\frac{\frac{\frac{-3}{2}}{b}}{3} \le 5.7824824102194 \cdot 10^{-107}:\\ \;\;\;\;\frac{-2}{a} \cdot \frac{b}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c}}{a \cdot 3}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if (/ (/ -3/2 b) 3) < -1.048690060768616e-86

    1. Initial program 31.9

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

    3. Applied *-un-lft-identity31.9

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

    4. Applied times-frac31.9

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

    5. Applied simplify31.9

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

    7. Applied flip--32.0

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

    8. Applied simplify16.8

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

    if -1.048690060768616e-86 < (/ (/ -3/2 b) 3) < 6.096801289818052e-306

    1. Initial program 58.0

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

    2. Taylor expanded around inf 15.0

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

    3. Applied simplify3.6

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} \cdot \frac{\frac{-3}{2}}{3}\right)}\]

    if 6.096801289818052e-306 < (/ (/ -3/2 b) 3) < 5.7824824102194e-107

    1. Initial program 46.2

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

    3. Applied *-un-lft-identity46.2

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

    4. Applied times-frac46.3

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

    5. Applied simplify46.3

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

    6. Taylor expanded around -inf 5.0

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\left(-2 \cdot \frac{b}{a}\right)}\]

    7. Applied simplify4.8

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

    if 5.7824824102194e-107 < (/ (/ -3/2 b) 3)

    1. Initial program 9.1

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
  3. Recombined 4 regimes into one program.

  4. Applied simplify9.4

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{-3}{2}}{b}}{3} \le -1.048690060768616 \cdot 10^{-86}:\\ \;\;\;\;\frac{-1}{3} \cdot \frac{\frac{a \cdot \left(3 \cdot c\right)}{b + \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 3}}}{a}\\ \mathbf{if}\;\frac{\frac{\frac{-3}{2}}{b}}{3} \le 6.096801289818052 \cdot 10^{-306}:\\ \;\;\;\;\frac{\frac{-3}{2}}{3} \cdot \frac{c}{b}\\ \mathbf{if}\;\frac{\frac{\frac{-3}{2}}{b}}{3} \le 5.7824824102194 \cdot 10^{-107}:\\ \;\;\;\;\frac{-2}{a} \cdot \frac{b}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c}}{a \cdot 3}\\ \end{array}}\]

Runtime

Time bar (total: 2.6m)Debug logProfile

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