\[\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a}\]
Test:
NMSE problem 3.2.1, negative
Bits:
128 bits
Bits error versus a
Bits error versus b/2
Bits error versus c
Time: 18.2 s
Input Error: 17.0
Output Error: 3.7
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{c}{\frac{\frac{1}{2} \cdot c}{\frac{b/2}{a}} - 2 \cdot b/2} & \text{when } b/2 \le -0.00030340048f0 \\ \frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a} & \text{when } b/2 \le 5.979008f+15 \\ \frac{(\left(\frac{c}{b/2} \cdot a\right) * \frac{1}{2} + \left(\left(-b/2\right) - b/2\right))_*}{a} & \text{otherwise} \end{cases}\)

    if b/2 < -0.00030340048f0

    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
      \[\frac{\frac{\color{red}{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\frac{\color{blue}{a \cdot c}}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}{a}\]
      15.4
    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
      \[\color{red}{\frac{\frac{a \cdot c}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - 2 \cdot b/2}}{a}} \leadsto \color{blue}{\frac{c}{\left(\frac{1}{2} \cdot c\right) \cdot \frac{a}{b/2} - 2 \cdot b/2}}\]
      1.5
    8. Applied taylor to get
      \[\frac{c}{\left(\frac{1}{2} \cdot c\right) \cdot \frac{a}{b/2} - 2 \cdot b/2} \leadsto \frac{c}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - 2 \cdot b/2}\]
      3.8
    9. Taylor expanded around 0 to get
      \[\frac{c}{\color{red}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2}} - 2 \cdot b/2} \leadsto \frac{c}{\color{blue}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2}} - 2 \cdot b/2}\]
      3.8
    10. Applied simplify to get
      \[\frac{c}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - 2 \cdot b/2} \leadsto \frac{c}{\frac{\frac{1}{2} \cdot c}{\frac{b/2}{a}} - 2 \cdot b/2}\]
      1.5
    11. Applied final simplification

    if -0.00030340048f0 < b/2 < 5.979008f+15

    1. Started with
      \[\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      6.2

    if 5.979008f+15 < b/2

    1. Started with
      \[\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      24.5
    2. Using strategy rm
      24.5
    3. Applied add-exp-log to get
      \[\frac{\left(-b/2\right) - \color{red}{\sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\left(-b/2\right) - \color{blue}{e^{\log \left(\sqrt{{b/2}^2 - a \cdot c}\right)}}}{a}\]
      24.9
    4. Applied taylor to get
      \[\frac{\left(-b/2\right) - e^{\log \left(\sqrt{{b/2}^2 - a \cdot c}\right)}}{a} \leadsto \frac{\left(-b/2\right) - e^{\log \left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}{a}\]
      8.6
    5. Taylor expanded around inf to get
      \[\frac{\left(-b/2\right) - e^{\log \color{red}{\left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}}{a} \leadsto \frac{\left(-b/2\right) - e^{\log \color{blue}{\left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}}{a}\]
      8.6
    6. Applied simplify to get
      \[\frac{\left(-b/2\right) - e^{\log \left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}{a} \leadsto \frac{\frac{\frac{1}{2} \cdot c}{\frac{b/2}{a}} + \left(\left(-b/2\right) - b/2\right)}{a}\]
      1.2
    7. Applied final simplification
    8. Applied simplify to get
      \[\color{red}{\frac{\frac{\frac{1}{2} \cdot c}{\frac{b/2}{a}} + \left(\left(-b/2\right) - b/2\right)}{a}} \leadsto \color{blue}{\frac{(\left(\frac{c}{b/2} \cdot a\right) * \frac{1}{2} + \left(\left(-b/2\right) - b/2\right))_*}{a}}\]
      1.3
  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))