\[\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a}\]
Test:
NMSE problem 3.2.1
Bits:
128 bits
Bits error versus a
Bits error versus b/2
Bits error versus c
Time: 14.6 s
Input Error: 17.8
Output Error: 2.3
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{b/2 + \left(-b/2\right)}{a} - \frac{1}{2} \cdot \frac{c}{b/2} & \text{when } b/2 \le -3.2880472f-13 \\ \frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a} & \text{when } b/2 \le 1.5915723f+15 \\ \frac{c}{\frac{b/2}{\frac{1}{2}}} - \frac{2 \cdot b/2}{a} & \text{otherwise} \end{cases}\)

    if b/2 < -3.2880472f-13

    1. Started with
      \[\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      28.0
    2. Applied taylor to get
      \[\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a} \leadsto \frac{\left(-b/2\right) - \left(\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - b/2\right)}{a}\]
      18.3
    3. Taylor expanded around -inf to get
      \[\frac{\left(-b/2\right) - \color{red}{\left(\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - b/2\right)}}{a} \leadsto \frac{\left(-b/2\right) - \color{blue}{\left(\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - b/2\right)}}{a}\]
      18.3
    4. Applied simplify to get
      \[\color{red}{\frac{\left(-b/2\right) - \left(\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - b/2\right)}{a}} \leadsto \color{blue}{\frac{b/2 + \left(-b/2\right)}{a} - \frac{1}{2} \cdot \frac{c}{b/2}}\]
      0.0

    if -3.2880472f-13 < b/2 < 1.5915723f+15

    1. Started with
      \[\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      5.3

    if 1.5915723f+15 < b/2

    1. Started with
      \[\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      24.1
    2. Applied taylor to get
      \[\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a} \leadsto \frac{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - 2 \cdot b/2}{a}\]
      6.2
    3. Taylor expanded around inf to get
      \[\frac{\color{red}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - 2 \cdot b/2}}{a} \leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - 2 \cdot b/2}}{a}\]
      6.2
    4. Applied simplify to get
      \[\color{red}{\frac{\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - 2 \cdot b/2}{a}} \leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{b/2}{c}} - \frac{b/2}{a} \cdot 2}\]
      0.0
    5. Applied taylor to get
      \[\frac{\frac{1}{2}}{\frac{b/2}{c}} - \frac{b/2}{a} \cdot 2 \leadsto \frac{1}{2} \cdot \frac{c}{b/2} - \frac{b/2}{a} \cdot 2\]
      0.0
    6. Taylor expanded around 0 to get
      \[\color{red}{\frac{1}{2} \cdot \frac{c}{b/2}} - \frac{b/2}{a} \cdot 2 \leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b/2}} - \frac{b/2}{a} \cdot 2\]
      0.0
    7. Applied simplify to get
      \[\frac{1}{2} \cdot \frac{c}{b/2} - \frac{b/2}{a} \cdot 2 \leadsto \frac{c}{\frac{b/2}{\frac{1}{2}}} - \frac{2 \cdot b/2}{a}\]
      0.3
    8. Applied final simplification
  1. Removed slow pow expressions

Original test:


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