\[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
Test:
NMSE problem 3.2.1, positive
Bits:
128 bits
Bits error versus a
Bits error versus b/2
Bits error versus c
Time: 14.4 s
Input Error: 36.1
Output Error: 4.3
Log: ⚲
Profile: 🕒
\(\begin{cases} -2 \cdot \frac{b/2}{a} & \text{when } b/2 \le -1.4548518266586696 \cdot 10^{-17} \\ \frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a} & \text{when } b/2 \le 1.0428439038139104 \cdot 10^{-180} \\ \frac{1}{\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{c}} & \text{when } b/2 \le 2.0626415808173556 \cdot 10^{+74} \\ \frac{b/2 + \left(-b/2\right)}{a} - \frac{c}{\frac{b/2}{\frac{1}{2}}} & \text{otherwise} \end{cases}\)

    if b/2 < -1.4548518266586696e-17

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      37.7
    2. Applied taylor to get
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a} \leadsto -2 \cdot \frac{b/2}{a}\]
      0.0
    3. Taylor expanded around -inf to get
      \[\color{red}{-2 \cdot \frac{b/2}{a}} \leadsto \color{blue}{-2 \cdot \frac{b/2}{a}}\]
      0.0

    if -1.4548518266586696e-17 < b/2 < 1.0428439038139104e-180

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      11.9

    if 1.0428439038139104e-180 < b/2 < 2.0626415808173556e+74

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      36.4
    2. Using strategy rm
      36.4
    3. Applied flip-+ to get
      \[\frac{\color{red}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\color{blue}{\frac{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}}{a}\]
      36.5
    4. Applied simplify to get
      \[\frac{\frac{\color{red}{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\frac{\color{blue}{a \cdot c}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a}\]
      16.7
    5. Using strategy rm
      16.7
    6. Applied clear-num to get
      \[\color{red}{\frac{\frac{a \cdot c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a}} \leadsto \color{blue}{\frac{1}{\frac{a}{\frac{a \cdot c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}}}\]
      16.8
    7. Applied simplify to get
      \[\frac{1}{\color{red}{\frac{a}{\frac{a \cdot c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}}} \leadsto \frac{1}{\color{blue}{\frac{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - c \cdot a}}{c}}}\]
      6.3
    8. Applied simplify to get
      \[\frac{1}{\frac{\color{red}{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - c \cdot a}}}{c}} \leadsto \frac{1}{\frac{\color{blue}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{c}}\]
      6.3

    if 2.0626415808173556e+74 < b/2

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      58.0
    2. Applied taylor to get
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a} \leadsto \frac{\left(-b/2\right) + \left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}{a}\]
      40.4
    3. Taylor expanded around inf to get
      \[\frac{\left(-b/2\right) + \color{red}{\left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}{a} \leadsto \frac{\left(-b/2\right) + \color{blue}{\left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}{a}\]
      40.4
    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
  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))