\[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
Test:
jeff quadratic root 1
Bits:
128 bits
Bits error versus a
Bits error versus b
Bits error versus c
Time: 14.5 s
Input Error: 18.4
Output Error: 3.9
Log: ⚲
Profile: 🕒
\(\begin{cases} \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + {\left(\sqrt{\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}\right)}^2} & \text{otherwise} \end{cases} & \text{when } {b}^2 - \left(4 \cdot a\right) \cdot c \le 1.8071351413641762 \cdot 10^{+300} \\ \frac{(\left(\frac{2 \cdot c}{b}\right) * a + \left(\left(-b\right) - b\right))_*}{a \cdot 2} & \text{when } b \ge 0 \\ \frac{c}{a \cdot \frac{c}{b} - b} & \text{otherwise} \end{cases}\)

    if (- (sqr b) (* (* 4 a) c)) < 1.8071351413641762e+300

    1. Started with
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      2.6
    2. Using strategy rm
      2.6
    3. Applied add-sqr-sqrt to get
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{red}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{blue}{\left(-b\right) + {\left(\sqrt{\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}\right)}^2}} & \text{otherwise} \end{cases}\]
      2.8

    if 1.8071351413641762e+300 < (- (sqr b) (* (* 4 a) c))

    1. Started with
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      46.7
    2. Applied taylor to get
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \left(b - 2 \cdot \frac{c \cdot a}{b}\right)}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      30.4
    3. Taylor expanded around inf to get
      \[\begin{cases} \frac{\left(-b\right) - \color{red}{\left(b - 2 \cdot \frac{c \cdot a}{b}\right)}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \color{blue}{\left(b - 2 \cdot \frac{c \cdot a}{b}\right)}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      30.4
    4. Applied simplify to get
      \[\color{red}{\begin{cases} \frac{\left(-b\right) - \left(b - 2 \cdot \frac{c \cdot a}{b}\right)}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}} \leadsto \color{blue}{\begin{cases} \frac{(\left(\frac{c}{\frac{b}{2}}\right) * a + \left(\left(-b\right) - b\right))_*}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)} + \left(-b\right)} & \text{otherwise} \end{cases}}\]
      26.6
    5. Applied taylor to get
      \[\begin{cases} \frac{(\left(\frac{c}{\frac{b}{2}}\right) * a + \left(\left(-b\right) - b\right))_*}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)} + \left(-b\right)} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{(\left(\frac{c}{\frac{b}{2}}\right) * a + \left(\left(-b\right) - b\right))_*}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b} & \text{otherwise} \end{cases}\]
      9.8
    6. Taylor expanded around -inf to get
      \[\begin{cases} \frac{(\left(\frac{c}{\frac{b}{2}}\right) * a + \left(\left(-b\right) - b\right))_*}{2 \cdot a} & \text{when } b \ge 0 \\ \color{red}{\frac{2 \cdot c}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{(\left(\frac{c}{\frac{b}{2}}\right) * a + \left(\left(-b\right) - b\right))_*}{2 \cdot a} & \text{when } b \ge 0 \\ \color{blue}{\frac{2 \cdot c}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}} & \text{otherwise} \end{cases}\]
      9.8
    7. Applied simplify to get
      \[\color{red}{\begin{cases} \frac{(\left(\frac{c}{\frac{b}{2}}\right) * a + \left(\left(-b\right) - b\right))_*}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b} & \text{otherwise} \end{cases}} \leadsto \color{blue}{\begin{cases} \frac{(\left(\frac{2 \cdot c}{b}\right) * a + \left(\left(-b\right) - b\right))_*}{a \cdot 2} & \text{when } b \ge 0 \\ \frac{c}{a \cdot \frac{c}{b} - b} & \text{otherwise} \end{cases}}\]
      5.8
  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)))))))