\[\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a}\]
Test:
NMSE problem 3.2.1, negative
Bits:
128 bits
Bits error versus a
Bits error versus b/2
Bits error versus c
Time: 1.5 m
Input Error: 34.3
Output Error: 6.3
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{c}{\left(a \cdot \frac{1}{2}\right) \cdot \frac{c}{b/2} - \left(b/2 - \left(-b/2\right)\right)} & \text{when } b/2 \le -5.5367438047953165 \cdot 10^{+140} \\ \frac{1}{\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - c \cdot a}}{c}} & \text{when } b/2 \le -5.4085235260168585 \cdot 10^{-306} \\ \frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a} & \text{when } b/2 \le 3.4791896352684183 \cdot 10^{+99} \\ \frac{\frac{\frac{1}{2} \cdot c}{\frac{b/2}{a}} + \left(\left(-b/2\right) - b/2\right)}{a} & \text{otherwise} \end{cases}\)

    if b/2 < -5.5367438047953165e+140

    1. Started with
      \[\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      61.8
    2. Using strategy rm
      61.8
    3. Applied flip-- to get
      \[\frac{\color{red}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\color{blue}{\frac{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}}{a}\]
      61.9
    4. Applied simplify to get
      \[\frac{\frac{\color{red}{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\frac{\color{blue}{a \cdot c}}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}{a}\]
      36.8
    5. Applied taylor to get
      \[\frac{\frac{a \cdot c}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\frac{a \cdot c}{\left(-b/2\right) + \left(\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - b/2\right)}}{a}\]
      14.3
    6. Taylor expanded around -inf to get
      \[\frac{\frac{a \cdot c}{\left(-b/2\right) + \color{red}{\left(\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - b/2\right)}}}{a} \leadsto \frac{\frac{a \cdot c}{\left(-b/2\right) + \color{blue}{\left(\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - b/2\right)}}}{a}\]
      14.3
    7. Applied simplify to get
      \[\color{red}{\frac{\frac{a \cdot c}{\left(-b/2\right) + \left(\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - b/2\right)}}{a}} \leadsto \color{blue}{\frac{c}{\left(a \cdot \frac{1}{2}\right) \cdot \frac{c}{b/2} - \left(b/2 - \left(-b/2\right)\right)}}\]
      0.8

    if -5.5367438047953165e+140 < b/2 < -5.4085235260168585e-306

    1. Started with
      \[\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      34.8
    2. Using strategy rm
      34.8
    3. Applied flip-- to get
      \[\frac{\color{red}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\color{blue}{\frac{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}}{a}\]
      34.9
    4. Applied simplify to get
      \[\frac{\frac{\color{red}{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\frac{\color{blue}{a \cdot c}}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}{a}\]
      16.6
    5. Using strategy rm
      16.6
    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
      \[\frac{1}{\color{red}{\frac{a}{\frac{a \cdot c}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}}} \leadsto \frac{1}{\color{blue}{\frac{\sqrt{b/2 \cdot b/2 - c \cdot a} + \left(-b/2\right)}{c}}}\]
      9.1
    8. Applied simplify to get
      \[\frac{1}{\frac{\color{red}{\sqrt{b/2 \cdot b/2 - c \cdot a} + \left(-b/2\right)}}{c}} \leadsto \frac{1}{\frac{\color{blue}{\left(-b/2\right) + \sqrt{{b/2}^2 - c \cdot a}}}{c}}\]
      9.1

    if -5.4085235260168585e-306 < b/2 < 3.4791896352684183e+99

    1. Started with
      \[\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      8.9

    if 3.4791896352684183e+99 < b/2

    1. Started with
      \[\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      46.5
    2. Applied taylor to get
      \[\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a} \leadsto \frac{\left(-b/2\right) - \left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}{a}\]
      11.4
    3. Taylor expanded around inf to get
      \[\frac{\left(-b/2\right) - \color{red}{\left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}{a} \leadsto \frac{\left(-b/2\right) - \color{blue}{\left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}{a}\]
      11.4
    4. Applied simplify to get
      \[\frac{\left(-b/2\right) - \left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}{a} \leadsto \frac{\frac{\frac{1}{2} \cdot c}{\frac{b/2}{a}} + \left(\left(-b/2\right) - b/2\right)}{a}\]
      2.1
    5. Applied final simplification
  1. Removed slow pow expressions

Original test:


(lambda ((a default) (b/2 default) (c default))
  #:name "NMSE problem 3.2.1, negative"
  (/ (- (- b/2) (sqrt (- (sqr b/2) (* a c)))) a))