\[\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a}\]
Test:
NMSE problem 3.2.1, negative
Bits:
128 bits
Bits error versus a
Bits error versus b/2
Bits error versus c
Time: 1.6 m
Input Error: 38.2
Output Error: 7.4
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{c}{\left(\frac{1}{2} \cdot c\right) \cdot \frac{a}{b/2} - 2 \cdot b/2} & \text{when } b/2 \le -5.030326967317398 \cdot 10^{-87} \\ \frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a} & \text{when } b/2 \le 3.4791896352684183 \cdot 10^{+99} \\ \frac{(\left(\frac{\frac{1}{2} \cdot c}{b/2}\right) * a + \left(\left(-b/2\right) - b/2\right))_*}{a} & \text{otherwise} \end{cases}\)

    if b/2 < -5.030326967317398e-87

    1. Started with
      \[\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      59.1
    2. Using strategy rm
      59.1
    3. Applied flip-- to get
      \[\frac{\color{red}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\color{blue}{\frac{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}}{a}\]
      59.1
    4. Applied simplify to get
      \[\frac{\frac{\color{red}{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\frac{\color{blue}{a \cdot c}}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}{a}\]
      35.1
    5. Applied taylor to get
      \[\frac{\frac{a \cdot c}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\frac{a \cdot c}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - 2 \cdot b/2}}{a}\]
      15.5
    6. Taylor expanded around -inf to get
      \[\frac{\frac{a \cdot c}{\color{red}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - 2 \cdot b/2}}}{a} \leadsto \frac{\frac{a \cdot c}{\color{blue}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - 2 \cdot b/2}}}{a}\]
      15.5
    7. Applied simplify to get
      \[\color{red}{\frac{\frac{a \cdot c}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - 2 \cdot b/2}}{a}} \leadsto \color{blue}{\frac{c}{\left(\frac{1}{2} \cdot c\right) \cdot \frac{a}{b/2} - 2 \cdot b/2}}\]
      4.1

    if -5.030326967317398e-87 < b/2 < 3.4791896352684183e+99

    1. Started with
      \[\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      13.0

    if 3.4791896352684183e+99 < b/2

    1. Started with
      \[\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      46.8
    2. Applied taylor to get
      \[\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a} \leadsto \frac{\left(-b/2\right) - \left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}{a}\]
      11.6
    3. Taylor expanded around inf to get
      \[\frac{\left(-b/2\right) - \color{red}{\left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}{a} \leadsto \frac{\left(-b/2\right) - \color{blue}{\left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}{a}\]
      11.6
    4. Applied simplify to get
      \[\color{red}{\frac{\left(-b/2\right) - \left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}{a}} \leadsto \color{blue}{\frac{(\left(\frac{\frac{1}{2} \cdot c}{b/2}\right) * a + \left(\left(-b/2\right) - b/2\right))_*}{a}}\]
      2.0
  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))