\[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
Test:
jeff quadratic root 1
Bits:
128 bits
Bits error versus a
Bits error versus b
Bits error versus c
Time: 23.5 s
Input Error: 21.0
Output Error: 5.3
Log: ⚲
Profile: 🕒
\(\begin{cases} \begin{cases} \frac{b}{c} - \frac{a}{b} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\frac{2 \cdot a}{\frac{b}{c}} - \left(b - \left(-b\right)\right)} & \text{otherwise} \end{cases} & \text{when } b \le -6.636085511448982 \cdot 10^{+62} \\ \begin{cases} \frac{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} & \text{when } b \le 2.0187374307498356 \cdot 10^{-297} \\ \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}} & \text{otherwise} \end{cases} & \text{when } b \le 2.7032921376893094 \cdot 10^{+83} \\ \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{when } b \ge 0 \\ \frac{1}{\frac{\frac{2 \cdot a}{\frac{b}{c}} - \left(b - \left(-b\right)\right)}{c \cdot 2}} & \text{otherwise} \end{cases}\)

    if b < -6.636085511448982e+62

    1. Started with
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      29.5
    2. Applied taylor to get
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases}\]
      7.8
    3. Taylor expanded around -inf to get
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{red}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)}} & \text{otherwise} \end{cases}\]
      7.8
    4. Applied taylor to get
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases}\]
      7.8
    5. Taylor expanded around inf to get
      \[\begin{cases} \frac{\color{red}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\color{blue}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases}\]
      7.8
    6. Applied simplify to get
      \[\color{red}{\begin{cases} \frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases}} \leadsto \color{blue}{\begin{cases} \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\frac{2 \cdot a}{\frac{b}{c}} - \left(b - \left(-b\right)\right)} & \text{otherwise} \end{cases}}\]
      1.5
    7. Using strategy rm
      1.5
    8. Applied add-cube-cbrt to get
      \[\begin{cases} \color{red}{\frac{\frac{c}{b}}{1} - \frac{b}{a}} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\frac{2 \cdot a}{\frac{b}{c}} - \left(b - \left(-b\right)\right)} & \text{otherwise} \end{cases} \leadsto \begin{cases} \color{blue}{{\left(\sqrt[3]{\frac{\frac{c}{b}}{1} - \frac{b}{a}}\right)}^3} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\frac{2 \cdot a}{\frac{b}{c}} - \left(b - \left(-b\right)\right)} & \text{otherwise} \end{cases}\]
      1.5
    9. Applied simplify to get
      \[\begin{cases} {\color{red}{\left(\sqrt[3]{\frac{\frac{c}{b}}{1} - \frac{b}{a}}\right)}}^3 & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\frac{2 \cdot a}{\frac{b}{c}} - \left(b - \left(-b\right)\right)} & \text{otherwise} \end{cases} \leadsto \begin{cases} {\color{blue}{\left(\sqrt[3]{\frac{c}{b} - \frac{b}{a}}\right)}}^3 & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\frac{2 \cdot a}{\frac{b}{c}} - \left(b - \left(-b\right)\right)} & \text{otherwise} \end{cases}\]
      1.5
    10. Applied taylor to get
      \[\begin{cases} {\left(\sqrt[3]{\frac{c}{b} - \frac{b}{a}}\right)}^3 & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\frac{2 \cdot a}{\frac{b}{c}} - \left(b - \left(-b\right)\right)} & \text{otherwise} \end{cases} \leadsto \begin{cases} {\left(\sqrt[3]{\frac{b}{c} - \frac{a}{b}}\right)}^3 & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\frac{2 \cdot a}{\frac{b}{c}} - \left(b - \left(-b\right)\right)} & \text{otherwise} \end{cases}\]
      1.5
    11. Taylor expanded around inf to get
      \[\begin{cases} {\color{red}{\left(\sqrt[3]{\frac{b}{c} - \frac{a}{b}}\right)}}^3 & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\frac{2 \cdot a}{\frac{b}{c}} - \left(b - \left(-b\right)\right)} & \text{otherwise} \end{cases} \leadsto \begin{cases} {\color{blue}{\left(\sqrt[3]{\frac{b}{c} - \frac{a}{b}}\right)}}^3 & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\frac{2 \cdot a}{\frac{b}{c}} - \left(b - \left(-b\right)\right)} & \text{otherwise} \end{cases}\]
      1.5
    12. Applied simplify to get
      \[\begin{cases} {\left(\sqrt[3]{\frac{b}{c} - \frac{a}{b}}\right)}^3 & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\frac{2 \cdot a}{\frac{b}{c}} - \left(b - \left(-b\right)\right)} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{b}{c} - \frac{a}{b} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\frac{2 \cdot a}{\frac{b}{c}} - \left(b - \left(-b\right)\right)} & \text{otherwise} \end{cases}\]
      1.5
    13. Applied final simplification

    if -6.636085511448982e+62 < b < 2.0187374307498356e-297

    1. Started with
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      9.6
    2. Using strategy rm
      9.6
    3. Applied flip-- to get
      \[\begin{cases} \frac{\color{red}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      9.6
    4. Applied simplify to get
      \[\begin{cases} \frac{\frac{\color{red}{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\frac{\color{blue}{\left(4 \cdot a\right) \cdot c}}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      9.6

    if 2.0187374307498356e-297 < b < 2.7032921376893094e+83

    1. Started with
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      8.4
    2. Using strategy rm
      8.4
    3. Applied flip-+ to get
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \color{red}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \color{blue}{\frac{2 \cdot c}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}} & \text{otherwise} \end{cases}\]
      8.4
    4. Applied simplify to get
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{\color{red}{2 \cdot c}}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{\color{blue}{2 \cdot c}}{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}} & \text{otherwise} \end{cases}\]
      8.4

    if 2.7032921376893094e+83 < b

    1. Started with
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      43.5
    2. Applied taylor to get
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases}\]
      43.5
    3. Taylor expanded around -inf to get
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{red}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)}} & \text{otherwise} \end{cases}\]
      43.5
    4. Applied taylor to get
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases}\]
      11.7
    5. Taylor expanded around inf to get
      \[\begin{cases} \frac{\color{red}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\color{blue}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases}\]
      11.7
    6. Applied simplify to get
      \[\color{red}{\begin{cases} \frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases}} \leadsto \color{blue}{\begin{cases} \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\frac{2 \cdot a}{\frac{b}{c}} - \left(b - \left(-b\right)\right)} & \text{otherwise} \end{cases}}\]
      0.0
    7. Using strategy rm
      0.0
    8. Applied clear-num to get
      \[\begin{cases} \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\frac{2 \cdot a}{\frac{b}{c}} - \left(b - \left(-b\right)\right)} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{when } b \ge 0 \\ \frac{1}{\frac{\frac{2 \cdot a}{\frac{b}{c}} - \left(b - \left(-b\right)\right)}{c \cdot 2}} & \text{otherwise} \end{cases}\]
      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)))))))