\[\begin{cases} \frac{2 \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{otherwise} \end{cases}\]
Test:
jeff quadratic root 2
Bits:
128 bits
Bits error versus a
Bits error versus b
Bits error versus c
Time: 17.3 s
Input Error: 40.7
Output Error: 3.0
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{c}{\frac{a}{b} \cdot c - b} & \text{when } b \ge 0 \\ \frac{\frac{c}{b}}{1} - \frac{\frac{b}{2}}{\frac{a}{2}} & \text{otherwise} \end{cases}\)
  1. Started with
    \[\begin{cases} \frac{2 \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{otherwise} \end{cases}\]
    40.7
  2. Applied taylor to get
    \[\begin{cases} \frac{2 \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{2 \cdot c}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{otherwise} \end{cases}\]
    23.4
  3. Taylor expanded around inf to get
    \[\begin{cases} \frac{2 \cdot c}{\color{red}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{2 \cdot c}{\color{blue}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{otherwise} \end{cases}\]
    23.4
  4. Applied simplify to get
    \[\color{red}{\begin{cases} \frac{2 \cdot c}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b} & \text{when } b \ge 0 \\ \frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{otherwise} \end{cases}} \leadsto \color{blue}{\begin{cases} \frac{c}{\frac{c}{b} \cdot a - b} & \text{when } b \ge 0 \\ \frac{\sqrt{{b}^2 - \left(c \cdot a\right) \cdot 4} + \left(-b\right)}{a \cdot 2} & \text{otherwise} \end{cases}}\]
    22.7
  5. Using strategy rm
    22.7
  6. Applied add-cube-cbrt to get
    \[\begin{cases} \color{red}{\frac{c}{\frac{c}{b} \cdot a - b}} & \text{when } b \ge 0 \\ \frac{\sqrt{{b}^2 - \left(c \cdot a\right) \cdot 4} + \left(-b\right)}{a \cdot 2} & \text{otherwise} \end{cases} \leadsto \begin{cases} \color{blue}{{\left(\sqrt[3]{\frac{c}{\frac{c}{b} \cdot a - b}}\right)}^3} & \text{when } b \ge 0 \\ \frac{\sqrt{{b}^2 - \left(c \cdot a\right) \cdot 4} + \left(-b\right)}{a \cdot 2} & \text{otherwise} \end{cases}\]
    23.2
  7. Using strategy rm
    23.2
  8. Applied add-cbrt-cube to get
    \[\begin{cases} {\color{red}{\left(\sqrt[3]{\frac{c}{\frac{c}{b} \cdot a - b}}\right)}}^3 & \text{when } b \ge 0 \\ \frac{\sqrt{{b}^2 - \left(c \cdot a\right) \cdot 4} + \left(-b\right)}{a \cdot 2} & \text{otherwise} \end{cases} \leadsto \begin{cases} {\color{blue}{\left(\sqrt[3]{{\left(\sqrt[3]{\frac{c}{\frac{c}{b} \cdot a - b}}\right)}^3}\right)}}^3 & \text{when } b \ge 0 \\ \frac{\sqrt{{b}^2 - \left(c \cdot a\right) \cdot 4} + \left(-b\right)}{a \cdot 2} & \text{otherwise} \end{cases}\]
    23.3
  9. Applied simplify to get
    \[\begin{cases} {\left(\sqrt[3]{\color{red}{{\left(\sqrt[3]{\frac{c}{\frac{c}{b} \cdot a - b}}\right)}^3}}\right)}^3 & \text{when } b \ge 0 \\ \frac{\sqrt{{b}^2 - \left(c \cdot a\right) \cdot 4} + \left(-b\right)}{a \cdot 2} & \text{otherwise} \end{cases} \leadsto \begin{cases} {\left(\sqrt[3]{\color{blue}{\frac{c}{\frac{c \cdot a}{b} - b}}}\right)}^3 & \text{when } b \ge 0 \\ \frac{\sqrt{{b}^2 - \left(c \cdot a\right) \cdot 4} + \left(-b\right)}{a \cdot 2} & \text{otherwise} \end{cases}\]
    23.8
  10. Applied taylor to get
    \[\begin{cases} {\left(\sqrt[3]{\frac{c}{\frac{c \cdot a}{b} - b}}\right)}^3 & \text{when } b \ge 0 \\ \frac{\sqrt{{b}^2 - \left(c \cdot a\right) \cdot 4} + \left(-b\right)}{a \cdot 2} & \text{otherwise} \end{cases} \leadsto \begin{cases} {\left(\sqrt[3]{\frac{c}{\frac{c \cdot a}{b} - b}}\right)}^3 & \text{when } b \ge 0 \\ \frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{a \cdot 2} & \text{otherwise} \end{cases}\]
    10.3
  11. Taylor expanded around -inf to get
    \[\begin{cases} {\left(\sqrt[3]{\frac{c}{\frac{c \cdot a}{b} - b}}\right)}^3 & \text{when } b \ge 0 \\ \frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{a \cdot 2} & \text{otherwise} \end{cases} \leadsto \begin{cases} {\left(\sqrt[3]{\frac{c}{\frac{c \cdot a}{b} - b}}\right)}^3 & \text{when } b \ge 0 \\ \frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{a \cdot 2} & \text{otherwise} \end{cases}\]
    10.3
  12. Applied simplify to get
    \[\begin{cases} {\left(\sqrt[3]{\frac{c}{\frac{c \cdot a}{b} - b}}\right)}^3 & \text{when } b \ge 0 \\ \frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{a \cdot 2} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{c}{a \cdot \frac{c}{b} - b} & \text{when } b \ge 0 \\ \frac{2}{a} \cdot \left(\frac{\frac{c}{b}}{\frac{2}{a}} - \frac{b}{2}\right) & \text{otherwise} \end{cases}\]
    7.2
  13. Applied final simplification
  14. Applied simplify to get
    \[\color{red}{\begin{cases} \frac{c}{a \cdot \frac{c}{b} - b} & \text{when } b \ge 0 \\ \frac{2}{a} \cdot \left(\frac{\frac{c}{b}}{\frac{2}{a}} - \frac{b}{2}\right) & \text{otherwise} \end{cases}} \leadsto \color{blue}{\begin{cases} \frac{c}{\frac{a}{b} \cdot c - b} & \text{when } b \ge 0 \\ \frac{\frac{c}{b}}{1} - \frac{\frac{b}{2}}{\frac{a}{2}} & \text{otherwise} \end{cases}}\]
    3.0

Original test:


(lambda ((a default) (b default) (c default))
  #:name "jeff quadratic root 2"
  (if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (- (sqr b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (sqr b) (* (* 4 a) c)))) (* 2 a))))