\[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
Test:
NMSE problem 3.2.1, positive
Bits:
128 bits
Bits error versus a
Bits error versus b/2
Bits error versus c
Time: 14.8 s
Input Error: 33.5
Output Error: 29.1
Log: ⚲
Profile: 🕒
\({\left(\frac{c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}\right)}^{1}\)
  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 simplify to get
    \[\frac{\frac{\color{red}{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\frac{\color{blue}{a \cdot c}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a}\]
    32.0
  5. Using strategy rm
    32.0
  6. Applied div-inv to get
    \[\color{red}{\frac{\frac{a \cdot c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a}} \leadsto \color{blue}{\frac{a \cdot c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}} \cdot \frac{1}{a}}\]
    32.1
  7. Using strategy rm
    32.1
  8. Applied pow1 to get
    \[\frac{a \cdot c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}} \cdot \color{red}{\frac{1}{a}} \leadsto \frac{a \cdot c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}} \cdot \color{blue}{{\left(\frac{1}{a}\right)}^{1}}\]
    32.1
  9. Applied pow1 to get
    \[\color{red}{\frac{a \cdot c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}} \cdot {\left(\frac{1}{a}\right)}^{1} \leadsto \color{blue}{{\left(\frac{a \cdot c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}\right)}^{1}} \cdot {\left(\frac{1}{a}\right)}^{1}\]
    32.1
  10. Applied pow-prod-down to get
    \[\color{red}{{\left(\frac{a \cdot c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}\right)}^{1} \cdot {\left(\frac{1}{a}\right)}^{1}} \leadsto \color{blue}{{\left(\frac{a \cdot c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}} \cdot \frac{1}{a}\right)}^{1}}\]
    32.1
  11. Applied simplify to get
    \[{\color{red}{\left(\frac{a \cdot c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}} \cdot \frac{1}{a}\right)}}^{1} \leadsto {\color{blue}{\left(\frac{c}{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - c \cdot a}}\right)}}^{1}\]
    29.1
  12. Applied simplify to get
    \[{\left(\frac{c}{\color{red}{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - c \cdot a}}}\right)}^{1} \leadsto {\left(\frac{c}{\color{blue}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}\right)}^{1}\]
    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))