\[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
Test:
jeff quadratic root 1
Bits:
128 bits
Bits error versus a
Bits error versus b
Bits error versus c
Time: 17.5 s
Input Error: 10.5
Output Error: 2.9
Log: ⚲
Profile: 🕒
\(\begin{cases} \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\frac{c \cdot 2}{\frac{b}{a}} - \left(b - \left(-b\right)\right)} & \text{otherwise} \end{cases} & \text{when } b \le -776.7439f0 \\ \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{{\left(\left(-b\right) + {\left(\sqrt[3]{\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}\right)}^3\right)}^{1}} & \text{otherwise} \end{cases} & \text{when } b \le 1.7973414f+17 \\ \frac{c}{b} - \frac{b}{a} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\left(-b\right) + \sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a}} & \text{otherwise} \end{cases}\)

    if b < -776.7439f0

    1. Started with
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      14.1
    2. Using strategy rm
      14.1
    3. Applied add-cube-cbrt to get
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{red}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{blue}{\left(-b\right) + {\left(\sqrt[3]{\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}\right)}^3}} & \text{otherwise} \end{cases}\]
      14.2
    4. Applied taylor to get
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + {\left(\sqrt[3]{\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}\right)}^3} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + {\left(\sqrt[3]{2 \cdot \frac{c \cdot a}{b} - b}\right)}^3} & \text{otherwise} \end{cases}\]
      3.7
    5. Taylor expanded around -inf to get
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-\color{red}{b}\right) + {\left(\sqrt[3]{2 \cdot \frac{c \cdot a}{b} - b}\right)}^3} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-\color{blue}{b}\right) + {\left(\sqrt[3]{2 \cdot \frac{c \cdot a}{b} - b}\right)}^3} & \text{otherwise} \end{cases}\]
      3.7
    6. Applied simplify to get
      \[\color{red}{\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + {\left(\sqrt[3]{2 \cdot \frac{c \cdot a}{b} - b}\right)}^3} & \text{otherwise} \end{cases}} \leadsto \color{blue}{\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\frac{c \cdot 2}{\frac{b}{a}} - \left(b - \left(-b\right)\right)} & \text{otherwise} \end{cases}}\]
      1.3

    if -776.7439f0 < b < 1.7973414f+17

    1. Started with
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      4.4
    2. Using strategy rm
      4.4
    3. Applied add-cube-cbrt to get
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{red}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{blue}{\left(-b\right) + {\left(\sqrt[3]{\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}\right)}^3}} & \text{otherwise} \end{cases}\]
      4.5
    4. Using strategy rm
      4.5
    5. Applied pow1 to get
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \color{red}{\frac{2 \cdot c}{\left(-b\right) + {\left(\sqrt[3]{\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}\right)}^3}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \color{blue}{\frac{2 \cdot c}{{\left(\left(-b\right) + {\left(\sqrt[3]{\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}\right)}^3\right)}^{1}}} & \text{otherwise} \end{cases}\]
      4.6

    if 1.7973414f+17 < b

    1. Started with
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      26.2
    2. Applied taylor to get
      \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      6.5
    3. Taylor expanded around inf to get
      \[\begin{cases} \frac{\color{red}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\color{blue}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
      6.5
    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. Applied simplify to get
      \[\begin{cases} \color{red}{\frac{\frac{c}{b}}{1} - \frac{b}{a}} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\left(-b\right) + \sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \color{blue}{\frac{c}{b} - \frac{b}{a}} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{\left(-b\right) + \sqrt{{b}^2 - \left(c \cdot 4\right) \cdot a}} & \text{otherwise} \end{cases}\]
      0.0
  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)))))))