Average Error: 44.0 → 12.3
Time: 16.7s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;c \le 247294268766.76083:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \mathbf{elif}\;c \le 8493198922459.606:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b} \cdot \sqrt[3]{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}\right) \cdot \sqrt[3]{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{\frac{\frac{a \cdot c}{b} \cdot \frac{-3}{2}}{3}}{a} \cdot \frac{\frac{\frac{a \cdot c}{b} \cdot \frac{-3}{2}}{3}}{a}\right) \cdot \frac{\frac{\frac{a \cdot c}{b} \cdot \frac{-3}{2}}{3}}{a}}\\ \end{array}\]

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 c < 247294268766.76083

    1. Initial program 45.1

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

    2. Simplified45.1

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

    3. Taylor expanded around inf 11.1

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

    if 247294268766.76083 < c < 8493198922459.606

    1. Initial program 34.1

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

    2. Simplified34.1

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

    4. Applied add-cube-cbrt34.1

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

    if 8493198922459.606 < c

    1. Initial program 31.5

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

    2. Simplified31.5

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

    3. Taylor expanded around inf 21.1

      \[\leadsto \frac{\color{blue}{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]
    4. Using strategy rm

    5. Applied associate-/r*21.0

      \[\leadsto \color{blue}{\frac{\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}{3}}{a}}\]
    6. Using strategy rm

    7. Applied add-cbrt-cube21.1

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

  4. Final simplification12.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;c \le 247294268766.76083:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \mathbf{elif}\;c \le 8493198922459.606:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b} \cdot \sqrt[3]{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}\right) \cdot \sqrt[3]{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{\frac{\frac{a \cdot c}{b} \cdot \frac{-3}{2}}{3}}{a} \cdot \frac{\frac{\frac{a \cdot c}{b} \cdot \frac{-3}{2}}{3}}{a}\right) \cdot \frac{\frac{\frac{a \cdot c}{b} \cdot \frac{-3}{2}}{3}}{a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019089 
(FPCore (a b c d)
  :name "Cubic critical, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))