\[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
Test:
NMSE problem 3.2.1, positive
Bits:
128 bits
Bits error versus a
Bits error versus b/2
Bits error versus c
Time: 15.1 s
Input Error: 34.7
Output Error: 6.7
Log: ⚲
Profile: 🕒
\(\begin{cases} -2 \cdot \frac{b/2}{a} & \text{when } b/2 \le -1.0526638788074079 \cdot 10^{+49} \\ \frac{1}{\frac{a}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}} & \text{when } b/2 \le 1.4445095675228828 \cdot 10^{-303} \\ \frac{\frac{a \cdot c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a} & \text{when } b/2 \le 1.3336442321324732 \cdot 10^{+29} \\ \frac{c}{\frac{a \cdot \frac{1}{2}}{\frac{b/2}{c}} + \left(\left(-b/2\right) - b/2\right)} & \text{otherwise} \end{cases}\)

    if b/2 < -1.0526638788074079e+49

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      41.1
    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 -1.0526638788074079e+49 < b/2 < 1.4445095675228828e-303

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      8.9
    2. Using strategy rm
      8.9
    3. Applied clear-num to get
      \[\color{red}{\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}} \leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}}\]
      9.0

    if 1.4445095675228828e-303 < b/2 < 1.3336442321324732e+29

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      26.1
    2. Using strategy rm
      26.1
    3. Applied flip-+ to get
      \[\frac{\color{red}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\color{blue}{\frac{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}}{a}\]
      26.2
    4. Applied simplify to get
      \[\frac{\frac{\color{red}{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\frac{\color{blue}{a \cdot c}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a}\]
      16.2

    if 1.3336442321324732e+29 < b/2

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      58.2
    2. Using strategy rm
      58.2
    3. Applied flip-+ to get
      \[\frac{\color{red}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\color{blue}{\frac{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}}{a}\]
      58.2
    4. Applied simplify to get
      \[\frac{\frac{\color{red}{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\frac{\color{blue}{a \cdot c}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a}\]
      31.6
    5. Applied taylor to get
      \[\frac{\frac{a \cdot c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\frac{a \cdot c}{\left(-b/2\right) - \left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}{a}\]
      14.8
    6. Taylor expanded around inf to get
      \[\frac{\frac{a \cdot c}{\left(-b/2\right) - \color{red}{\left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}}{a} \leadsto \frac{\frac{a \cdot c}{\left(-b/2\right) - \color{blue}{\left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}}{a}\]
      14.8
    7. Applied simplify to get
      \[\color{red}{\frac{\frac{a \cdot c}{\left(-b/2\right) - \left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}{a}} \leadsto \color{blue}{\frac{c}{\frac{a \cdot \frac{1}{2}}{\frac{b/2}{c}} + \left(\left(-b/2\right) - b/2\right)}}\]
      2.1
  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))