\[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
Test:
jeff quadratic root 1
Bits:
128 bits
Bits error versus a
Bits error versus b
Bits error versus c
Time: 22.4 s
Input Error: 21.9
Output Error: 4.6
Log: ⚲
Profile: 🕒
\(\begin{cases} \begin{cases} \frac{c}{b} - \frac{b}{a} & \text{when } b \ge 0 \\ \frac{-c}{b} & \text{otherwise} \end{cases} & \text{when } b \le -2.2913476789857995 \cdot 10^{+35} \\ \begin{cases} \frac{b}{a} \cdot \frac{-2}{2} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}} & \text{otherwise} \end{cases} & \text{when } b \le 8.227588915381728 \cdot 10^{-306} \\ \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases} & \text{when } b \le 1.0887108629649695 \cdot 10^{+34} \\ \frac{c}{b} - \frac{b}{a} & \text{when } b \ge 0 \\ \frac{\frac{2}{4}}{a} \cdot \left(\left(\left(-b\right) + b\right) - \left(2 \cdot a\right) \cdot \frac{c}{b}\right) & \text{otherwise} \end{cases}\)

    if b < -2.2913476789857995e+35

    1. Started with
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      27.8
    2. Applied taylor to get
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      27.8
    3. Taylor expanded around inf to get
      \[\begin{cases} \frac{\color{red}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\color{blue}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      27.8
    4. Applied simplify to get
      \[\color{red}{\begin{cases} \frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}} \leadsto \color{blue}{\begin{cases} \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\left(-b\right) + \sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a}} & \text{otherwise} \end{cases}}\]
      27.7
    5. Using strategy rm
      27.7
    6. Applied clear-num to get
      \[\begin{cases} \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\left(-b\right) + \sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{when } b \ge 0 \\ \frac{1}{\frac{\left(-b\right) + \sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a}}{c \cdot 2}} & \text{otherwise} \end{cases}\]
      28.1
    7. Applied taylor to get
      \[\begin{cases} \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{when } b \ge 0 \\ \frac{1}{\frac{\left(-b\right) + \sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a}}{c \cdot 2}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{when } b \ge 0 \\ -1 \cdot \frac{c}{b} & \text{otherwise} \end{cases}\]
      0
    8. Taylor expanded around -inf to get
      \[\begin{cases} \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{when } b \ge 0 \\ -1 \cdot \frac{c}{b} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{when } b \ge 0 \\ -1 \cdot \frac{c}{b} & \text{otherwise} \end{cases}\]
      0
    9. Applied simplify to get
      \[\begin{cases} \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{when } b \ge 0 \\ -1 \cdot \frac{c}{b} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{c}{b} - \frac{b}{a} & \text{when } b \ge 0 \\ \frac{-c}{b} & \text{otherwise} \end{cases}\]
      0
    10. Applied final simplification

    if -2.2913476789857995e+35 < b < 8.227588915381728e-306

    1. Started with
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      10.8
    2. Using strategy rm
      10.8
    3. Applied flip-- to get
      \[\begin{cases} \frac{\color{red}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      10.8
    4. Applied simplify to get
      \[\begin{cases} \frac{\frac{\color{red}{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\frac{\color{blue}{\left(4 \cdot a\right) \cdot c}}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      10.8
    5. Applied taylor to get
      \[\begin{cases} \frac{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{-2 \cdot b}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      9.9
    6. Taylor expanded around 0 to get
      \[\begin{cases} \frac{\color{red}{-2 \cdot b}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\color{blue}{-2 \cdot b}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      9.9
    7. Applied simplify to get
      \[\begin{cases} \frac{-2 \cdot b}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{b}{a} \cdot \frac{-2}{2} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}} & \text{otherwise} \end{cases}\]
      9.9
    8. Applied final simplification

    if 8.227588915381728e-306 < b < 1.0887108629649695e+34

    1. Started with
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      9.7
    2. Applied taylor to get
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases}\]
      9.7
    3. Taylor expanded around -inf to get
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{red}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)}} & \text{otherwise} \end{cases}\]
      9.7

    if 1.0887108629649695e+34 < b

    1. Started with
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      38.7
    2. Applied taylor to get
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      10.1
    3. Taylor expanded around inf to get
      \[\begin{cases} \frac{\color{red}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\color{blue}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      10.1
    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. Using strategy rm
      0.0
    6. Applied flip-+ to get
      \[\begin{cases} \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{when } b \ge 0 \\ \color{red}{\frac{c \cdot 2}{\left(-b\right) + \sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a}}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{when } b \ge 0 \\ \color{blue}{\frac{c \cdot 2}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a}\right)}^2}{\left(-b\right) - \sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a}}}} & \text{otherwise} \end{cases}\]
      0.0
    7. Applied associate-/r/ to get
      \[\begin{cases} \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a}\right)}^2}{\left(-b\right) - \sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a}}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a}\right)}^2} \cdot \left(\left(-b\right) - \sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a}\right) & \text{otherwise} \end{cases}\]
      0.0
    8. Applied simplify to get
      \[\begin{cases} \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a}\right)}^2} \cdot \left(\left(-b\right) - \sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a}\right) & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{when } b \ge 0 \\ \frac{\frac{2 \cdot c}{4}}{a \cdot c} \cdot \left(\left(-b\right) - \sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a}\right) & \text{otherwise} \end{cases}\]
      0.0
    9. Applied taylor to get
      \[\begin{cases} \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{when } b \ge 0 \\ \frac{\frac{2 \cdot c}{4}}{a \cdot c} \cdot \left(\left(-b\right) - \sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a}\right) & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{when } b \ge 0 \\ \frac{\frac{2 \cdot c}{4}}{a \cdot c} \cdot \left(\left(-b\right) - \left(2 \cdot \frac{c \cdot a}{b} - b\right)\right) & \text{otherwise} \end{cases}\]
      0.0
    10. Taylor expanded around -inf to get
      \[\begin{cases} \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{when } b \ge 0 \\ \frac{\frac{2 \cdot c}{4}}{a \cdot c} \cdot \color{red}{\left(\left(-b\right) - \left(2 \cdot \frac{c \cdot a}{b} - b\right)\right)} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{when } b \ge 0 \\ \frac{\frac{2 \cdot c}{4}}{a \cdot c} \cdot \color{blue}{\left(\left(-b\right) - \left(2 \cdot \frac{c \cdot a}{b} - b\right)\right)} & \text{otherwise} \end{cases}\]
      0.0
    11. Applied simplify to get
      \[\begin{cases} \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{when } b \ge 0 \\ \frac{\frac{2 \cdot c}{4}}{a \cdot c} \cdot \left(\left(-b\right) - \left(2 \cdot \frac{c \cdot a}{b} - b\right)\right) & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{c}{b} - \frac{b}{a} & \text{when } b \ge 0 \\ \left(\frac{\frac{2}{4}}{a} \cdot \frac{c}{c}\right) \cdot \left(\left(b + \left(-b\right)\right) - \frac{c \cdot 2}{\frac{b}{a}}\right) & \text{otherwise} \end{cases}\]
      0.0
    12. Applied final simplification
    13. Applied simplify to get
      \[\color{red}{\begin{cases} \frac{c}{b} - \frac{b}{a} & \text{when } b \ge 0 \\ \left(\frac{\frac{2}{4}}{a} \cdot \frac{c}{c}\right) \cdot \left(\left(b + \left(-b\right)\right) - \frac{c \cdot 2}{\frac{b}{a}}\right) & \text{otherwise} \end{cases}} \leadsto \color{blue}{\begin{cases} \frac{c}{b} - \frac{b}{a} & \text{when } b \ge 0 \\ \frac{\frac{2}{4}}{a} \cdot \left(\left(\left(-b\right) + b\right) - \left(2 \cdot a\right) \cdot \frac{c}{b}\right) & \text{otherwise} \end{cases}}\]
      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)))))))