\[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
Test:
NMSE problem 3.2.1, positive
Bits:
128 bits
Bits error versus a
Bits error versus b/2
Bits error versus c
Time: 12.4 s
Input Error: 33.5
Output Error: 29.1
Log: ⚲
Profile: 🕒
\(\frac{c}{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - c \cdot a}}\)
  1. Started with
    \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
    33.5
  2. Using strategy rm
    33.5
  3. Applied flip-+ to get
    \[\frac{\color{red}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\color{blue}{\frac{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}}{a}\]
    43.2
  4. Applied associate-/l/ to get
    \[\color{red}{\frac{\frac{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a}} \leadsto \color{blue}{\frac{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}{a \cdot \left(\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}\right)}}\]
    45.8
  5. Applied taylor to get
    \[\frac{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}{a \cdot \left(\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}\right)} \leadsto \frac{c \cdot a}{a \cdot \left(\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}\right)}\]
    34.6
  6. Taylor expanded around inf to get
    \[\frac{\color{red}{c \cdot a}}{a \cdot \left(\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}\right)} \leadsto \frac{\color{blue}{c \cdot a}}{a \cdot \left(\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}\right)}\]
    34.6
  7. Applied simplify to get
    \[\frac{c \cdot a}{a \cdot \left(\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}\right)} \leadsto \frac{\frac{c}{1}}{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c}}\]
    29.1
  8. Applied final simplification
  9. Applied simplify to get
    \[\color{red}{\frac{\frac{c}{1}}{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c}}} \leadsto \color{blue}{\frac{c}{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - c \cdot a}}}\]
    29.1

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))