\[\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a}\]
Test:
NMSE problem 3.2.1, negative
Bits:
128 bits
Bits error versus a
Bits error versus b/2
Bits error versus c
Time: 14.4 s
Input Error: 16.9
Output Error: 4.1
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{c}{\frac{\frac{1}{2} \cdot c}{\frac{b/2}{a}} - 2 \cdot b/2} & \text{when } b/2 \le -0.15658416f0 \\ \frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a} & \text{when } b/2 \le 1.458361f+14 \\ \frac{-2}{\frac{a}{b/2}} & \text{otherwise} \end{cases}\)

    if b/2 < -0.15658416f0

    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
      \[\frac{\color{red}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\color{blue}{\frac{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}}{a}\]
      29.7
    4. Applied simplify to get
      \[\frac{\frac{\color{red}{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\frac{\color{blue}{a \cdot c}}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}{a}\]
      14.8
    5. Applied taylor to get
      \[\frac{\frac{a \cdot c}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\frac{a \cdot c}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - 2 \cdot b/2}}{a}\]
      7.3
    6. Taylor expanded around -inf to get
      \[\frac{\frac{a \cdot c}{\color{red}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - 2 \cdot b/2}}}{a} \leadsto \frac{\frac{a \cdot c}{\color{blue}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - 2 \cdot b/2}}}{a}\]
      7.3
    7. Applied simplify to get
      \[\color{red}{\frac{\frac{a \cdot c}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - 2 \cdot b/2}}{a}} \leadsto \color{blue}{\frac{c}{\left(\frac{1}{2} \cdot c\right) \cdot \frac{a}{b/2} - 2 \cdot b/2}}\]
      1.6
    8. Applied taylor to get
      \[\frac{c}{\left(\frac{1}{2} \cdot c\right) \cdot \frac{a}{b/2} - 2 \cdot b/2} \leadsto \frac{c}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - 2 \cdot b/2}\]
      4.0
    9. Taylor expanded around 0 to get
      \[\frac{c}{\color{red}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2}} - 2 \cdot b/2} \leadsto \frac{c}{\color{blue}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2}} - 2 \cdot b/2}\]
      4.0
    10. Applied simplify to get
      \[\frac{c}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - 2 \cdot b/2} \leadsto \frac{c}{\frac{\frac{1}{2} \cdot c}{\frac{b/2}{a}} - 2 \cdot b/2}\]
      1.6
    11. Applied final simplification

    if -0.15658416f0 < b/2 < 1.458361f+14

    1. Started with
      \[\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      6.9

    if 1.458361f+14 < b/2

    1. Started with
      \[\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      23.6
    2. Using strategy rm
      23.6
    3. Applied add-cube-cbrt to get
      \[\color{red}{\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a}} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a}}\right)}^3}\]
      23.7
    4. Applied taylor to get
      \[{\left(\sqrt[3]{\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a}}\right)}^3 \leadsto {\left(\sqrt[3]{-2 \cdot \frac{b/2}{a}}\right)}^3\]
      0.8
    5. Taylor expanded around inf to get
      \[{\left(\sqrt[3]{\color{red}{-2 \cdot \frac{b/2}{a}}}\right)}^3 \leadsto {\left(\sqrt[3]{\color{blue}{-2 \cdot \frac{b/2}{a}}}\right)}^3\]
      0.8
    6. Applied simplify to get
      \[{\left(\sqrt[3]{-2 \cdot \frac{b/2}{a}}\right)}^3 \leadsto \frac{-2}{\frac{a}{b/2}}\]
      0.3
    7. 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))