\[\begin{cases} \frac{2 \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{otherwise} \end{cases}\]
Test:
jeff quadratic root 2
Bits:
128 bits
Bits error versus a
Bits error versus b
Bits error versus c
Time: 15.8 s
Input Error: 10.0
Output Error: 2.7
Log: ⚲
Profile: 🕒
\(\begin{cases} \begin{cases} \frac{-2}{\frac{b}{c}} & \text{when } b \ge 0 \\ \frac{c}{b} - \frac{b}{a} & \text{otherwise} \end{cases} & \text{when } b \le -1.6510623f+14 \\ \begin{cases} \frac{2 \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + {\left(\sqrt{\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}\right)}^2}{2 \cdot a} & \text{otherwise} \end{cases} & \text{when } b \le 5.594203f+08 \\ \frac{c}{\frac{c}{\frac{b}{a}} - b} & \text{when } b \ge 0 \\ \frac{-b}{a} & \text{otherwise} \end{cases}\)

    if b < -1.6510623f+14

    1. Started with
      \[\begin{cases} \frac{2 \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{otherwise} \end{cases}\]
      24.0
    2. Applied taylor to get
      \[\begin{cases} \frac{2 \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{2 \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{when } b \ge 0 \\ \frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{2 \cdot a} & \text{otherwise} \end{cases}\]
      6.3
    3. Taylor expanded around -inf to get
      \[\begin{cases} \frac{2 \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{when } b \ge 0 \\ \frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{2 \cdot a} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{2 \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{when } b \ge 0 \\ \frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{2 \cdot a} & \text{otherwise} \end{cases}\]
      6.3
    4. Applied simplify to get
      \[\color{red}{\begin{cases} \frac{2 \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{when } b \ge 0 \\ \frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{2 \cdot a} & \text{otherwise} \end{cases}} \leadsto \color{blue}{\begin{cases} \frac{c \cdot 2}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot c\right) \cdot a}} & \text{when } b \ge 0 \\ \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{otherwise} \end{cases}}\]
      0.0
    5. Applied taylor to get
      \[\begin{cases} \frac{c \cdot 2}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot c\right) \cdot a}} & \text{when } b \ge 0 \\ \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{otherwise} \end{cases} \leadsto \begin{cases} -2 \cdot \frac{c}{b} & \text{when } b \ge 0 \\ \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{otherwise} \end{cases}\]
      0.0
    6. Taylor expanded around inf to get
      \[\begin{cases} \color{red}{-2 \cdot \frac{c}{b}} & \text{when } b \ge 0 \\ \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{otherwise} \end{cases} \leadsto \begin{cases} \color{blue}{-2 \cdot \frac{c}{b}} & \text{when } b \ge 0 \\ \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{otherwise} \end{cases}\]
      0.0
    7. Applied simplify to get
      \[\begin{cases} -2 \cdot \frac{c}{b} & \text{when } b \ge 0 \\ \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{-2}{\frac{b}{c}} & \text{when } b \ge 0 \\ \frac{c}{b} - \frac{b}{a} & \text{otherwise} \end{cases}\]
      0.0
    8. Applied final simplification

    if -1.6510623f+14 < b < 5.594203f+08

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

    if 5.594203f+08 < b

    1. Started with
      \[\begin{cases} \frac{2 \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{otherwise} \end{cases}\]
      14.7
    2. Applied taylor to get
      \[\begin{cases} \frac{2 \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{2 \cdot c}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{otherwise} \end{cases}\]
      3.7
    3. Taylor expanded around inf to get
      \[\begin{cases} \frac{2 \cdot c}{\color{red}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{2 \cdot c}{\color{blue}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{otherwise} \end{cases}\]
      3.7
    4. Applied simplify to get
      \[\color{red}{\begin{cases} \frac{2 \cdot c}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{otherwise} \end{cases}} \leadsto \color{blue}{\begin{cases} \frac{c}{\frac{c}{b} \cdot a - b} & \text{when } b \ge 0 \\ \frac{\sqrt{{b}^2 - \left(c \cdot a\right) \cdot 4} + \left(-b\right)}{a \cdot 2} & \text{otherwise} \end{cases}}\]
      0.8
    5. Applied taylor to get
      \[\begin{cases} \frac{c}{\frac{c}{b} \cdot a - b} & \text{when } b \ge 0 \\ \frac{\sqrt{{b}^2 - \left(c \cdot a\right) \cdot 4} + \left(-b\right)}{a \cdot 2} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{c}{\frac{c}{b} \cdot a - b} & \text{when } b \ge 0 \\ -1 \cdot \frac{b}{a} & \text{otherwise} \end{cases}\]
      0.8
    6. Taylor expanded around -inf to get
      \[\begin{cases} \frac{c}{\frac{c}{b} \cdot a - b} & \text{when } b \ge 0 \\ -1 \cdot \frac{b}{a} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{c}{\frac{c}{b} \cdot a - b} & \text{when } b \ge 0 \\ -1 \cdot \frac{b}{a} & \text{otherwise} \end{cases}\]
      0.8
    7. Applied simplify to get
      \[\begin{cases} \frac{c}{\frac{c}{b} \cdot a - b} & \text{when } b \ge 0 \\ -1 \cdot \frac{b}{a} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{c}{\frac{c}{\frac{b}{a}} - b} & \text{when } b \ge 0 \\ \frac{-b}{a} & \text{otherwise} \end{cases}\]
      0.8
    8. Applied final simplification
  1. Removed slow pow expressions

Original test:


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