\[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
Test:
NMSE problem 3.2.1, positive
Bits:
128 bits
Bits error versus a
Bits error versus b/2
Bits error versus c
Time: 12.4 s
Input Error: 34.9
Output Error: 7.1
Log: ⚲
Profile: 🕒
\(\begin{cases} -2 \cdot \frac{b/2}{a} & \text{when } b/2 \le -1.2339538201069979 \cdot 10^{+148} \\ \frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a} & \text{when } b/2 \le 4.6117267249984834 \cdot 10^{-185} \\ \frac{\frac{a \cdot c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a} & \text{when } b/2 \le 2.4608343160951844 \cdot 10^{+34} \\ \frac{b/2 + \left(-b/2\right)}{a} - \frac{c}{\frac{b/2}{\frac{1}{2}}} & \text{otherwise} \end{cases}\)

    if b/2 < -1.2339538201069979e+148

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      58.5
    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 -1.2339538201069979e+148 < b/2 < 4.6117267249984834e-185

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      10.4

    if 4.6117267249984834e-185 < b/2 < 2.4608343160951844e+34

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      34.0
    2. Using strategy rm
      34.0
    3. Applied flip-+ to get
      \[\frac{\color{red}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\color{blue}{\frac{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}}{a}\]
      34.1
    4. Applied simplify to get
      \[\frac{\frac{\color{red}{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\frac{\color{blue}{a \cdot c}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a}\]
      17.7

    if 2.4608343160951844e+34 < b/2

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      57.9
    2. Applied taylor to get
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a} \leadsto \frac{\left(-b/2\right) + \left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}{a}\]
      41.4
    3. Taylor expanded around inf to get
      \[\frac{\left(-b/2\right) + \color{red}{\left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}{a} \leadsto \frac{\left(-b/2\right) + \color{blue}{\left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}{a}\]
      41.4
    4. Applied simplify to get
      \[\color{red}{\frac{\left(-b/2\right) + \left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}{a}} \leadsto \color{blue}{\frac{b/2 + \left(-b/2\right)}{a} - \frac{c}{\frac{b/2}{\frac{1}{2}}}}\]
      0.0
  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))