Average Error: 33.5 → 12.3
Time: 27.6s
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.847241431798756 \cdot 10^{+152}:\\ \;\;\;\;\frac{-2}{3} \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 2.1697769394994967 \cdot 10^{-59}:\\ \;\;\;\;\frac{\sqrt{(\left(-3\right) \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*} - b}{a} \cdot \frac{1}{3}\\ \mathbf{elif}\;b \le 2.1112254168143557 \cdot 10^{+152}:\\ \;\;\;\;\frac{\left(a \cdot c\right) \cdot 3}{\left(\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}\right) \cdot \left(a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot c}{b} \cdot \frac{-3}{2}}{a \cdot 3}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Split input into 4 regimes

  2. if b < -6.847241431798756e+152

    1. Initial program 60.7

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

    2. Taylor expanded around -inf 2.7

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

    if -6.847241431798756e+152 < b < 2.1697769394994967e-59

    1. Initial program 12.6

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

    3. Applied *-un-lft-identity12.6

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

    4. Applied times-frac12.7

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

    5. Simplified12.7

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

    if 2.1697769394994967e-59 < b < 2.1112254168143557e+152

    1. Initial program 44.2

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

    3. Applied flip-+44.3

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

    4. Applied associate-/l/46.3

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

    5. Simplified15.5

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

    if 2.1112254168143557e+152 < b

    1. Initial program 62.8

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

    2. Taylor expanded around inf 13.6

      \[\leadsto \frac{\color{blue}{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification12.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.847241431798756 \cdot 10^{+152}:\\ \;\;\;\;\frac{-2}{3} \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 2.1697769394994967 \cdot 10^{-59}:\\ \;\;\;\;\frac{\sqrt{(\left(-3\right) \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*} - b}{a} \cdot \frac{1}{3}\\ \mathbf{elif}\;b \le 2.1112254168143557 \cdot 10^{+152}:\\ \;\;\;\;\frac{\left(a \cdot c\right) \cdot 3}{\left(\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}\right) \cdot \left(a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot c}{b} \cdot \frac{-3}{2}}{a \cdot 3}\\ \end{array}\]

Runtime

Time bar (total: 27.6s)Debug logProfile

herbie shell --seed 2018221 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))