\[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
Test:
NMSE problem 3.2.1, positive
Bits:
128 bits
Bits error versus a
Bits error versus b/2
Bits error versus c
Time: 12.7 s
Input Error: 17.6
Output Error: 2.7
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\frac{1}{2}}{\frac{b/2}{c}} - \frac{b/2}{a} \cdot 2 & \text{when } b/2 \le -1.3686002f-10 \\ \frac{\frac{a}{1} \cdot \frac{c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a} & \text{when } b/2 \le 1.7973414f+17 \\ \frac{c}{\frac{a \cdot \frac{1}{2}}{\frac{b/2}{c}} + \left(\left(-b/2\right) - b/2\right)} & \text{otherwise} \end{cases}\)

    if b/2 < -1.3686002f-10

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      17.2
    2. Applied taylor to get
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a} \leadsto \frac{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - 2 \cdot b/2}{a}\]
      4.5
    3. Taylor expanded around -inf to get
      \[\frac{\color{red}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - 2 \cdot b/2}}{a} \leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - 2 \cdot b/2}}{a}\]
      4.5
    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 -1.3686002f-10 < b/2 < 1.7973414f+17

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      12.4
    2. Using strategy rm
      12.4
    3. Applied flip-+ to get
      \[\frac{\color{red}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\color{blue}{\frac{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}}{a}\]
      14.6
    4. Applied simplify to get
      \[\frac{\frac{\color{red}{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\frac{\color{blue}{a \cdot c}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a}\]
      7.4
    5. Using strategy rm
      7.4
    6. Applied *-un-lft-identity to get
      \[\frac{\frac{a \cdot c}{\color{red}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}}{a} \leadsto \frac{\frac{a \cdot c}{\color{blue}{1 \cdot \left(\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}\right)}}}{a}\]
      7.4
    7. Applied times-frac to get
      \[\frac{\color{red}{\frac{a \cdot c}{1 \cdot \left(\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}\right)}}}{a} \leadsto \frac{\color{blue}{\frac{a}{1} \cdot \frac{c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}}{a}\]
      5.5

    if 1.7973414f+17 < b/2

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      30.0
    2. Using strategy rm
      30.0
    3. Applied flip-+ to get
      \[\frac{\color{red}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\color{blue}{\frac{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}}{a}\]
      30.8
    4. Applied simplify to get
      \[\frac{\frac{\color{red}{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\frac{\color{blue}{a \cdot c}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a}\]
      17.8
    5. Applied taylor to get
      \[\frac{\frac{a \cdot c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\frac{a \cdot c}{\left(-b/2\right) - \left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}{a}\]
      7.6
    6. Taylor expanded around inf to get
      \[\frac{\frac{a \cdot c}{\left(-b/2\right) - \color{red}{\left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}}{a} \leadsto \frac{\frac{a \cdot c}{\left(-b/2\right) - \color{blue}{\left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}}{a}\]
      7.6
    7. Applied simplify to get
      \[\color{red}{\frac{\frac{a \cdot c}{\left(-b/2\right) - \left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}{a}} \leadsto \color{blue}{\frac{c}{\frac{a \cdot \frac{1}{2}}{\frac{b/2}{c}} + \left(\left(-b/2\right) - b/2\right)}}\]
      0.8
  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))