\[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
Test:
NMSE problem 3.2.1, positive
Bits:
128 bits
Bits error versus a
Bits error versus b/2
Bits error versus c
Time: 13.6 s
Input Error: 17.7
Output Error: 3.0
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\frac{1}{2}}{\frac{b/2}{c}} - \frac{b/2}{a} \cdot 2 & \text{when } b/2 \le -1.3462214f+10 \\ \frac{{\left(\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}\right)}^{1}}{a} & \text{when } b/2 \le 1.75507f-11 \\ \frac{c}{\frac{\frac{1}{2} \cdot c}{\frac{b/2}{a}} + \left(\left(-b/2\right) - b/2\right)} & \text{otherwise} \end{cases}\)

    if b/2 < -1.3462214f+10

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      20.8
    2. Applied taylor to get
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a} \leadsto \frac{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - 2 \cdot b/2}{a}\]
      5.8
    3. Taylor expanded around -inf to get
      \[\frac{\color{red}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - 2 \cdot b/2}}{a} \leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - 2 \cdot b/2}}{a}\]
      5.8
    4. Applied simplify to get
      \[\color{red}{\frac{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - 2 \cdot b/2}{a}} \leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{b/2}{c}} - \frac{b/2}{a} \cdot 2}\]
      0.1

    if -1.3462214f+10 < b/2 < 1.75507f-11

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      5.5
    2. Using strategy rm
      5.5
    3. Applied pow1 to get
      \[\frac{\color{red}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\color{blue}{{\left(\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}\right)}^{1}}}{a}\]
      5.5

    if 1.75507f-11 < b/2

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      28.0
    2. Using strategy rm
      28.0
    3. Applied flip-+ to get
      \[\frac{\color{red}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\color{blue}{\frac{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}}{a}\]
      29.5
    4. Applied simplify to get
      \[\frac{\frac{\color{red}{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\frac{\color{blue}{a \cdot c}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a}\]
      16.5
    5. Applied taylor to get
      \[\frac{\frac{a \cdot c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\frac{a \cdot c}{\left(-b/2\right) - \left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}{a}\]
      7.4
    6. Taylor expanded around inf to get
      \[\frac{\frac{a \cdot c}{\left(-b/2\right) - \color{red}{\left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}}{a} \leadsto \frac{\frac{a \cdot c}{\left(-b/2\right) - \color{blue}{\left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}}{a}\]
      7.4
    7. Applied simplify to get
      \[\color{red}{\frac{\frac{a \cdot c}{\left(-b/2\right) - \left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}{a}} \leadsto \color{blue}{\frac{c}{\frac{a \cdot \frac{1}{2}}{\frac{b/2}{c}} + \left(\left(-b/2\right) - b/2\right)}}\]
      2.0
    8. Applied taylor to get
      \[\frac{c}{\frac{a \cdot \frac{1}{2}}{\frac{b/2}{c}} + \left(\left(-b/2\right) - b/2\right)} \leadsto \frac{c}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} + \left(\left(-b/2\right) - b/2\right)}\]
      4.0
    9. Taylor expanded around 0 to get
      \[\frac{c}{\color{red}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2}} + \left(\left(-b/2\right) - b/2\right)} \leadsto \frac{c}{\color{blue}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2}} + \left(\left(-b/2\right) - b/2\right)}\]
      4.0
    10. Applied simplify to get
      \[\frac{c}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} + \left(\left(-b/2\right) - b/2\right)} \leadsto \frac{c}{\frac{\frac{1}{2} \cdot c}{\frac{b/2}{a}} + \left(\left(-b/2\right) - b/2\right)}\]
      1.9
    11. Applied final simplification
  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))