Average Error: 33.6 → 15.2
Time: 28.6s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.3372661925740899 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{3}{2} \cdot \frac{a \cdot c}{b} - 2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \le 1.4624567676001702 \cdot 10^{-80}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}{a \cdot 3}\\ \end{array}\]

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 3 regimes

  2. if b < -1.3372661925740899e+154

    1. Initial program 60.8

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

    2. Taylor expanded around -inf 9.9

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

    if -1.3372661925740899e+154 < b < 1.4624567676001702e-80

    1. Initial program 11.7

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

    2. Initial simplification11.8

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

    4. Applied associate-*r*11.8

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

    if 1.4624567676001702e-80 < b

    1. Initial program 52.2

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

    2. Taylor expanded around inf 20.7

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

  4. Final simplification15.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3372661925740899 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{3}{2} \cdot \frac{a \cdot c}{b} - 2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \le 1.4624567676001702 \cdot 10^{-80}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}{a \cdot 3}\\ \end{array}\]

Runtime

Time bar (total: 28.6s)Debug logProfile

herbie shell --seed 2018346 
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))