\[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
Test:
NMSE problem 3.2.1, positive
Bits:
128 bits
Bits error versus a
Bits error versus b/2
Bits error versus c
Time: 27.2 s
Input Error: 33.6
Output Error: 5.7
Log: ⚲
Profile: 🕒
\(\begin{cases} -2 \cdot \frac{b/2}{a} & \text{when } b/2 \le -1.9805160440224734 \cdot 10^{+152} \\ \frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a} & \text{when } b/2 \le -2.5142405720522603 \cdot 10^{-272} \\ \frac{1}{\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{c}} & \text{when } b/2 \le 3.741572576046744 \cdot 10^{+123} \\ \frac{c}{\frac{a \cdot \frac{1}{2}}{\frac{b/2}{c}} + \left(\left(-b/2\right) - b/2\right)} & \text{otherwise} \end{cases}\)

    if b/2 < -1.9805160440224734e+152

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      60.1
    2. Applied taylor to get
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a} \leadsto -2 \cdot \frac{b/2}{a}\]
      0
    3. Taylor expanded around -inf to get
      \[\color{red}{-2 \cdot \frac{b/2}{a}} \leadsto \color{blue}{-2 \cdot \frac{b/2}{a}}\]
      0

    if -1.9805160440224734e+152 < b/2 < -2.5142405720522603e-272

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      7.4

    if -2.5142405720522603e-272 < b/2 < 3.741572576046744e+123

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      32.2
    2. Using strategy rm
      32.2
    3. Applied flip-+ to get
      \[\frac{\color{red}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\color{blue}{\frac{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}}{a}\]
      32.4
    4. Applied simplify to get
      \[\frac{\frac{\color{red}{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\frac{\color{blue}{a \cdot c}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a}\]
      15.1
    5. Using strategy rm
      15.1
    6. Applied clear-num to get
      \[\color{red}{\frac{\frac{a \cdot c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a}} \leadsto \color{blue}{\frac{1}{\frac{a}{\frac{a \cdot c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}}}\]
      15.3
    7. Applied simplify to get
      \[\frac{1}{\color{red}{\frac{a}{\frac{a \cdot c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}}} \leadsto \frac{1}{\color{blue}{\frac{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - c \cdot a}}{c}}}\]
      8.7
    8. Applied simplify to get
      \[\frac{1}{\frac{\color{red}{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - c \cdot a}}}{c}} \leadsto \frac{1}{\frac{\color{blue}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{c}}\]
      8.7

    if 3.741572576046744e+123 < b/2

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      60.8
    2. Using strategy rm
      60.8
    3. Applied flip-+ to get
      \[\frac{\color{red}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\color{blue}{\frac{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}}{a}\]
      60.8
    4. Applied simplify to get
      \[\frac{\frac{\color{red}{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\frac{\color{blue}{a \cdot c}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a}\]
      34.5
    5. Applied taylor to get
      \[\frac{\frac{a \cdot c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\frac{a \cdot c}{\left(-b/2\right) - \left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}{a}\]
      15.4
    6. Taylor expanded around inf to get
      \[\frac{\frac{a \cdot c}{\left(-b/2\right) - \color{red}{\left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}}{a} \leadsto \frac{\frac{a \cdot c}{\left(-b/2\right) - \color{blue}{\left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}}{a}\]
      15.4
    7. Applied simplify to get
      \[\color{red}{\frac{\frac{a \cdot c}{\left(-b/2\right) - \left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}{a}} \leadsto \color{blue}{\frac{c}{\frac{a \cdot \frac{1}{2}}{\frac{b/2}{c}} + \left(\left(-b/2\right) - b/2\right)}}\]
      1.1
  1. Removed slow pow expressions

Original test:


(lambda ((a default) (b/2 default) (c default))
  #:name "NMSE problem 3.2.1, positive"
  (/ (+ (- b/2) (sqrt (- (sqr b/2) (* a c)))) a))