\[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
Test:
jeff quadratic root 1
Bits:
128 bits
Bits error versus a
Bits error versus b
Bits error versus c
Time: 19.8 s
Input Error: 9.9
Output Error: 2.8
Log: ⚲
Profile: 🕒
\(\begin{cases} \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - a \cdot \left(c \cdot 4\right)}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\frac{c \cdot 2}{\frac{b}{a}} - \left(b - \left(-b\right)\right)} & \text{otherwise} \end{cases} & \text{when } b \le -1.2459677f+19 \\ \begin{cases} \frac{\left(-b\right) - {\left(\sqrt{\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}\right)}^2}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} & \text{when } b \le 9.210374f+08 \\ \frac{b}{a} \cdot \frac{-2}{2} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}} & \text{otherwise} \end{cases}\)

    if b < -1.2459677f+19

    1. Started with
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      18.3
    2. Using strategy rm
      18.3
    3. Applied add-sqr-sqrt to get
      \[\begin{cases} \frac{\left(-b\right) - \color{red}{\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \color{blue}{{\left(\sqrt{\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}\right)}^2}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      18.3
    4. Applied taylor to get
      \[\begin{cases} \frac{\left(-b\right) - {\left(\sqrt{\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}\right)}^2}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - {\left(\sqrt{\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}\right)}^2}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases}\]
      3.7
    5. Taylor expanded around -inf to get
      \[\begin{cases} \frac{\left(-b\right) - {\left(\sqrt{\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}\right)}^2}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{red}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - {\left(\sqrt{\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}\right)}^2}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)}} & \text{otherwise} \end{cases}\]
      3.7
    6. Applied simplify to get
      \[\begin{cases} \frac{\left(-b\right) - {\left(\sqrt{\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}\right)}^2}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - a \cdot \left(c \cdot 4\right)}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\frac{c \cdot 2}{\frac{b}{a}} - \left(b - \left(-b\right)\right)} & \text{otherwise} \end{cases}\]
      0.5
    7. Applied final simplification

    if -1.2459677f+19 < b < 9.210374f+08

    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
      \[\begin{cases} \frac{\left(-b\right) - \color{red}{\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \color{blue}{{\left(\sqrt{\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}\right)}^2}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      4.3

    if 9.210374f+08 < b

    1. Started with
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      21.3
    2. Using strategy rm
      21.3
    3. Applied flip-- to get
      \[\begin{cases} \frac{\color{red}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      29.8
    4. Applied simplify to get
      \[\begin{cases} \frac{\frac{\color{red}{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\frac{\color{blue}{\left(4 \cdot a\right) \cdot c}}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      30.0
    5. Applied taylor to get
      \[\begin{cases} \frac{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{-2 \cdot b}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      0.3
    6. Taylor expanded around 0 to get
      \[\begin{cases} \frac{\color{red}{-2 \cdot b}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\color{blue}{-2 \cdot b}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      0.3
    7. Applied simplify to get
      \[\begin{cases} \frac{-2 \cdot b}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{b}{a} \cdot \frac{-2}{2} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}} & \text{otherwise} \end{cases}\]
      0
    8. Applied final simplification
  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)))))))