\[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
Test:
jeff quadratic root 1
Bits:
128 bits
Bits error versus a
Bits error versus b
Bits error versus c
Time: 19.1 s
Input Error: 30.5
Output Error: 4.1
Log: ⚲
Profile: 🕒
\(\begin{cases} \begin{cases} \frac{c}{b} - \frac{b}{a} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\frac{2 \cdot a}{\frac{b}{c}} - \left(b - \left(-b\right)\right)} & \text{otherwise} \end{cases} & \text{when } {b}^2 - \left(4 \cdot a\right) \cdot c \le 1.8236611171008294 \cdot 10^{-299} \\ \begin{cases} \frac{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} & \text{when } {b}^2 - \left(4 \cdot a\right) \cdot c \le 4.734305638751365 \cdot 10^{-209} \\ \begin{cases} \frac{\left(-b\right) - \sqrt[3]{{\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^3}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} & \text{when } {b}^2 - \left(4 \cdot a\right) \cdot c \le 5.7333944931174024 \cdot 10^{+215} \\ \frac{c}{b} - \frac{b}{a} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\frac{2 \cdot a}{\frac{b}{c}} - \left(b - \left(-b\right)\right)} & \text{otherwise} \end{cases}\)

    if (- (sqr b) (* (* 4 a) c)) < 1.8236611171008294e-299 or 5.7333944931174024e+215 < (- (sqr b) (* (* 4 a) c))

    1. Started with
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      48.6
    2. Applied taylor to get
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases}\]
      28.2
    3. Taylor expanded around -inf to get
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{red}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)}} & \text{otherwise} \end{cases}\]
      28.2
    4. Applied taylor to get
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases}\]
      12.9
    5. Taylor expanded around inf to get
      \[\begin{cases} \frac{\color{red}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\color{blue}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases}\]
      12.9
    6. Applied simplify to get
      \[\color{red}{\begin{cases} \frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases}} \leadsto \color{blue}{\begin{cases} \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\frac{2 \cdot a}{\frac{b}{c}} - \left(b - \left(-b\right)\right)} & \text{otherwise} \end{cases}}\]
      5.1
    7. Applied simplify to get
      \[\begin{cases} \color{red}{\frac{\frac{c}{b}}{1} - \frac{b}{a}} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\frac{2 \cdot a}{\frac{b}{c}} - \left(b - \left(-b\right)\right)} & \text{otherwise} \end{cases} \leadsto \begin{cases} \color{blue}{\frac{c}{b} - \frac{b}{a}} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\frac{2 \cdot a}{\frac{b}{c}} - \left(b - \left(-b\right)\right)} & \text{otherwise} \end{cases}\]
      5.1

    if 1.8236611171008294e-299 < (- (sqr b) (* (* 4 a) c)) < 4.734305638751365e-209

    1. Started with
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      0.1
    2. Using strategy rm
      0.1
    3. Applied flip-- to get
      \[\begin{cases} \frac{\color{red}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      18.7
    4. Applied simplify to get
      \[\begin{cases} \frac{\frac{\color{red}{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\frac{\color{blue}{\left(4 \cdot a\right) \cdot c}}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      18.8

    if 4.734305638751365e-209 < (- (sqr b) (* (* 4 a) c)) < 5.7333944931174024e+215

    1. Started with
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      3.1
    2. Using strategy rm
      3.1
    3. Applied add-cbrt-cube to get
      \[\begin{cases} \frac{\left(-b\right) - \color{red}{\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \color{blue}{\sqrt[3]{{\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^3}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      4.1
  1. Removed slow pow expressions

Original test:


(lambda ((a default) (b default) (c default))
  #:name "jeff quadratic root 1"
  (if (>= b 0) (/ (- (- b) (sqrt (- (sqr b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (sqr b) (* (* 4 a) c)))))))