\[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
Test:
NMSE problem 3.2.1, positive
Bits:
128 bits
Bits error versus a
Bits error versus b/2
Bits error versus c
Time: 14.4 s
Input Error: 35.6
Output Error: 6.5
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{1}{2} \cdot \frac{c}{b/2} - 2 \cdot \frac{b/2}{a} & \text{when } b/2 \le -1.102640798064626 \cdot 10^{+96} \\ \frac{\left(-b/2\right) + {\left(\sqrt{\sqrt{{b/2}^2 - a \cdot c}}\right)}^2}{a} & \text{when } b/2 \le -4.180861961583105 \cdot 10^{-193} \\ \frac{\frac{a \cdot c}{a}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}} & \text{when } b/2 \le 2.2608845850534584 \cdot 10^{+29} \\ \frac{b/2 + \left(-b/2\right)}{a} - \frac{c}{\frac{b/2}{\frac{1}{2}}} & \text{otherwise} \end{cases}\)

    if b/2 < -1.102640798064626e+96

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      47.2
    2. Using strategy rm
      47.2
    3. Applied div-inv to get
      \[\color{red}{\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}} \leadsto \color{blue}{\left(\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}\right) \cdot \frac{1}{a}}\]
      47.2
    4. Applied taylor to get
      \[\left(\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}\right) \cdot \frac{1}{a} \leadsto \frac{1}{2} \cdot \frac{c}{b/2} - 2 \cdot \frac{b/2}{a}\]
      0.0
    5. Taylor expanded around -inf to get
      \[\color{red}{\frac{1}{2} \cdot \frac{c}{b/2} - 2 \cdot \frac{b/2}{a}} \leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b/2} - 2 \cdot \frac{b/2}{a}}\]
      0.0

    if -1.102640798064626e+96 < b/2 < -4.180861961583105e-193

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      7.2
    2. Using strategy rm
      7.2
    3. Applied add-sqr-sqrt to get
      \[\frac{\left(-b/2\right) + \color{red}{\sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\left(-b/2\right) + \color{blue}{{\left(\sqrt{\sqrt{{b/2}^2 - a \cdot c}}\right)}^2}}{a}\]
      7.4

    if -4.180861961583105e-193 < b/2 < 2.2608845850534584e+29

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      25.6
    2. Using strategy rm
      25.6
    3. Applied div-inv to get
      \[\color{red}{\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}} \leadsto \color{blue}{\left(\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}\right) \cdot \frac{1}{a}}\]
      25.6
    4. Using strategy rm
      25.6
    5. Applied flip-+ to get
      \[\color{red}{\left(\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}\right)} \cdot \frac{1}{a} \leadsto \color{blue}{\frac{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}} \cdot \frac{1}{a}\]
      25.9
    6. Applied associate-*l/ to get
      \[\color{red}{\frac{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}} \cdot \frac{1}{a}} \leadsto \color{blue}{\frac{\left({\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2\right) \cdot \frac{1}{a}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}\]
      26.0
    7. Applied simplify to get
      \[\frac{\color{red}{\left({\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2\right) \cdot \frac{1}{a}}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}} \leadsto \frac{\color{blue}{\frac{a \cdot c}{a}}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}\]
      16.5

    if 2.2608845850534584e+29 < b/2

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      58.1
    2. Applied taylor to get
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a} \leadsto \frac{\left(-b/2\right) + \left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}{a}\]
      41.9
    3. Taylor expanded around inf to get
      \[\frac{\left(-b/2\right) + \color{red}{\left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}{a} \leadsto \frac{\left(-b/2\right) + \color{blue}{\left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}{a}\]
      41.9
    4. Applied simplify to get
      \[\color{red}{\frac{\left(-b/2\right) + \left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}{a}} \leadsto \color{blue}{\frac{b/2 + \left(-b/2\right)}{a} - \frac{c}{\frac{b/2}{\frac{1}{2}}}}\]
      0.1
  1. Removed slow pow expressions

Original test:


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