\[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
Test:
jeff quadratic root 1
Bits:
128 bits
Bits error versus a
Bits error versus b
Bits error versus c
Time: 15.7 s
Input Error: 10.1
Output Error: 2.6
Log: ⚲
Profile: 🕒
\(\begin{cases} \begin{cases} \frac{b + \left(-b\right)}{a \cdot 2} - \frac{c}{b} & \text{when } b \ge 0 \\ \frac{c}{\frac{c}{b} \cdot a - b} & \text{otherwise} \end{cases} & \text{when } b \le -2.376507f+13 \\ \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + {\left(\sqrt{\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}\right)}^2} & \text{otherwise} \end{cases} & \text{when } b \le 100403176.0f0 \\ \frac{c}{b} - \frac{b}{a} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\left(-b\right) + \sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a}} & \text{otherwise} \end{cases}\)

    if b < -2.376507f+13

    1. Started with
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      15.9
    2. Applied taylor to get
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b} & \text{otherwise} \end{cases}\]
      4.0
    3. Taylor expanded around -inf to get
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \color{red}{\frac{2 \cdot c}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \color{blue}{\frac{2 \cdot c}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}} & \text{otherwise} \end{cases}\]
      4.0
    4. Applied simplify to get
      \[\color{red}{\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b} & \text{otherwise} \end{cases}} \leadsto \color{blue}{\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{c}{\frac{c \cdot a}{b} - b} & \text{otherwise} \end{cases}}\]
      3.8
    5. Applied taylor to get
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{c}{\frac{c \cdot a}{b} - b} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \left(2 \cdot \frac{c \cdot a}{b} - b\right)}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{c}{\frac{c \cdot a}{b} - b} & \text{otherwise} \end{cases}\]
      3.8
    6. Taylor expanded around -inf to get
      \[\begin{cases} \frac{\left(-b\right) - \color{red}{\left(2 \cdot \frac{c \cdot a}{b} - b\right)}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{c}{\frac{c \cdot a}{b} - b} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \color{blue}{\left(2 \cdot \frac{c \cdot a}{b} - b\right)}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{c}{\frac{c \cdot a}{b} - b} & \text{otherwise} \end{cases}\]
      3.8
    7. Applied simplify to get
      \[\color{red}{\begin{cases} \frac{\left(-b\right) - \left(2 \cdot \frac{c \cdot a}{b} - b\right)}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{c}{\frac{c \cdot a}{b} - b} & \text{otherwise} \end{cases}} \leadsto \color{blue}{\begin{cases} \frac{\left(-b\right) + b}{2 \cdot a} - \frac{\frac{c}{b}}{1} & \text{when } b \ge 0 \\ \frac{c}{\frac{c}{b} \cdot a - b} & \text{otherwise} \end{cases}}\]
      0.6
    8. Applied simplify to get
      \[\begin{cases} \color{red}{\frac{\left(-b\right) + b}{2 \cdot a} - \frac{\frac{c}{b}}{1}} & \text{when } b \ge 0 \\ \frac{c}{\frac{c}{b} \cdot a - b} & \text{otherwise} \end{cases} \leadsto \begin{cases} \color{blue}{\frac{b + \left(-b\right)}{a \cdot 2} - \frac{c}{b}} & \text{when } b \ge 0 \\ \frac{c}{\frac{c}{b} \cdot a - b} & \text{otherwise} \end{cases}\]
      0.6

    if -2.376507f+13 < b < 100403176.0f0

    1. Started with
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      4.2
    2. Using strategy rm
      4.2
    3. Applied add-sqr-sqrt to get
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{red}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{blue}{\left(-b\right) + {\left(\sqrt{\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}\right)}^2}} & \text{otherwise} \end{cases}\]
      4.3

    if 100403176.0f0 < b

    1. Started with
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      20.0
    2. Applied taylor to get
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      5.4
    3. Taylor expanded around inf to get
      \[\begin{cases} \frac{\color{red}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\color{blue}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      5.4
    4. Applied simplify to get
      \[\color{red}{\begin{cases} \frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}} \leadsto \color{blue}{\begin{cases} \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\left(-b\right) + \sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a}} & \text{otherwise} \end{cases}}\]
      0.0
    5. Applied simplify to get
      \[\begin{cases} \color{red}{\frac{\frac{c}{b}}{1} - \frac{b}{a}} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\left(-b\right) + \sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \color{blue}{\frac{c}{b} - \frac{b}{a}} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\left(-b\right) + \sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a}} & \text{otherwise} \end{cases}\]
      0.0
  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)))))))