Average Error: 33.4 → 10.3
Time: 27.1s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.3323489384795222 \cdot 10^{+151}:\\ \;\;\;\;(\frac{-2}{3} \cdot \left(\frac{b}{a}\right) + \left(\frac{c}{\frac{b}{\frac{1}{2}}}\right))_*\\ \mathbf{elif}\;b \le 1.1958712990443948 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{\sqrt{(\left(-3 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} + \left(-b\right)}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\ \end{array}\]

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 < -1.3323489384795222e+151

    1. Initial program 60.0

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

    2. Taylor expanded around inf 60.0

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

    3. Simplified60.0

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

    4. Taylor expanded around -inf 2.8

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b} - \frac{2}{3} \cdot \frac{b}{a}}\]

    5. Simplified2.8

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

    if -1.3323489384795222e+151 < b < 1.1958712990443948e-79

    1. Initial program 12.1

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

    2. Taylor expanded around inf 12.2

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

    3. Simplified12.1

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

    5. Applied associate-/r*12.2

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

    if 1.1958712990443948e-79 < b

    1. Initial program 51.5

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

    2. Taylor expanded around inf 51.5

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

    3. Simplified51.5

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

    5. Applied associate-/r*51.5

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

    6. Taylor expanded around inf 10.2

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3323489384795222 \cdot 10^{+151}:\\ \;\;\;\;(\frac{-2}{3} \cdot \left(\frac{b}{a}\right) + \left(\frac{c}{\frac{b}{\frac{1}{2}}}\right))_*\\ \mathbf{elif}\;b \le 1.1958712990443948 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{\sqrt{(\left(-3 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} + \left(-b\right)}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\ \end{array}\]

Runtime

Time bar (total: 27.1s)Debug logProfile

herbie shell --seed 2018273 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))