\[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
Test:
NMSE problem 3.2.1, positive
Bits:
128 bits
Bits error versus a
Bits error versus b/2
Bits error versus c
Time: 22.4 s
Input Error: 15.8
Output Error: 2.9
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\frac{1}{2}}{\frac{b/2}{c}} - \frac{b/2}{a} \cdot 2 & \text{when } b/2 \le -1.3809824f+19 \\ \frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a} & \text{when } b/2 \le 1.7689368f-34 \\ \frac{\frac{a}{1} \cdot \frac{c}{\left(-b/2\right) - e^{\log \left(\sqrt{{b/2}^2 - a \cdot c}\right)}}}{a} & \text{when } b/2 \le 3.318237f+16 \\ \frac{-1}{2} \cdot \frac{c}{b/2} & \text{otherwise} \end{cases}\)

    if b/2 < -1.3809824f+19

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      28.8
    2. Applied taylor to get
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a} \leadsto \frac{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - 2 \cdot b/2}{a}\]
      6.7
    3. Taylor expanded around -inf to get
      \[\frac{\color{red}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - 2 \cdot b/2}}{a} \leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - 2 \cdot b/2}}{a}\]
      6.7
    4. Applied simplify to get
      \[\color{red}{\frac{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - 2 \cdot b/2}{a}} \leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{b/2}{c}} - \frac{b/2}{a} \cdot 2}\]
      0.0

    if -1.3809824f+19 < b/2 < 1.7689368f-34

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      3.9

    if 1.7689368f-34 < b/2 < 3.318237f+16

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      15.5
    2. Using strategy rm
      15.5
    3. Applied flip-+ to get
      \[\frac{\color{red}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\color{blue}{\frac{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}}{a}\]
      17.0
    4. Applied simplify to get
      \[\frac{\frac{\color{red}{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\frac{\color{blue}{a \cdot c}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a}\]
      6.5
    5. Using strategy rm
      6.5
    6. Applied *-un-lft-identity to get
      \[\frac{\frac{a \cdot c}{\color{red}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}}{a} \leadsto \frac{\frac{a \cdot c}{\color{blue}{1 \cdot \left(\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}\right)}}}{a}\]
      6.5
    7. Applied times-frac to get
      \[\frac{\color{red}{\frac{a \cdot c}{1 \cdot \left(\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}\right)}}}{a} \leadsto \frac{\color{blue}{\frac{a}{1} \cdot \frac{c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}}{a}\]
      3.5
    8. Using strategy rm
      3.5
    9. Applied add-exp-log to get
      \[\frac{\frac{a}{1} \cdot \frac{c}{\left(-b/2\right) - \color{red}{\sqrt{{b/2}^2 - a \cdot c}}}}{a} \leadsto \frac{\frac{a}{1} \cdot \frac{c}{\left(-b/2\right) - \color{blue}{e^{\log \left(\sqrt{{b/2}^2 - a \cdot c}\right)}}}}{a}\]
      4.8

    if 3.318237f+16 < b/2

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      29.7
    2. Applied taylor to get
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a} \leadsto \frac{\frac{-1}{2} \cdot \frac{c \cdot a}{b/2}}{a}\]
      6.7
    3. Taylor expanded around inf to get
      \[\frac{\color{red}{\frac{-1}{2} \cdot \frac{c \cdot a}{b/2}}}{a} \leadsto \frac{\color{blue}{\frac{-1}{2} \cdot \frac{c \cdot a}{b/2}}}{a}\]
      6.7
    4. Applied simplify to get
      \[\color{red}{\frac{\frac{-1}{2} \cdot \frac{c \cdot a}{b/2}}{a}} \leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b/2}}\]
      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))