\[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
Test:
NMSE p42, negative
Bits:
128 bits
Bits error versus a
Bits error versus b
Bits error versus c
Time: 20.4 s
Input Error: 33.5
Output Error: 30.6
Log: ⚲
Profile: 🕒
\(\frac{1}{\left(\left(-b\right) + \sqrt{{b}^2 - \left(a \cdot c\right) \cdot 4}\right) \cdot \frac{\frac{2}{c}}{4}}\)
  1. Started with
    \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
    33.5
  2. Using strategy rm
    33.5
  3. Applied flip-- to get
    \[\frac{\color{red}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
    44.0
  4. Applied simplify to get
    \[\frac{\frac{\color{red}{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\frac{\color{blue}{c \cdot \left(4 \cdot a\right)}}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
    33.4
  5. Using strategy rm
    33.4
  6. Applied clear-num to get
    \[\color{red}{\frac{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}} \leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}}\]
    33.4
  7. Applied simplify to get
    \[\frac{1}{\color{red}{\frac{2 \cdot a}{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}} \leadsto \frac{1}{\color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right) \cdot \left(\frac{1}{c} \cdot \frac{2}{4}\right)}}\]
    30.6
  8. Applied simplify to get
    \[\frac{1}{\color{red}{\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)} \cdot \left(\frac{1}{c} \cdot \frac{2}{4}\right)} \leadsto \frac{1}{\color{blue}{\left(\left(-b\right) + \sqrt{{b}^2 - \left(a \cdot c\right) \cdot 4}\right)} \cdot \left(\frac{1}{c} \cdot \frac{2}{4}\right)}\]
    30.6
  9. Applied simplify to get
    \[\frac{1}{\left(\left(-b\right) + \sqrt{{b}^2 - \left(a \cdot c\right) \cdot 4}\right) \cdot \color{red}{\left(\frac{1}{c} \cdot \frac{2}{4}\right)}} \leadsto \frac{1}{\left(\left(-b\right) + \sqrt{{b}^2 - \left(a \cdot c\right) \cdot 4}\right) \cdot \color{blue}{\frac{\frac{2}{c}}{4}}}\]
    30.6

Original test:


(lambda ((a default) (b default) (c default))
  #:name "NMSE p42, negative"
  (/ (- (- b) (sqrt (- (sqr b) (* 4 (* a c))))) (* 2 a))
  #:target
  (if (< b 0) (/ c (* a (/ (+ (- b) (sqrt (- (sqr b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (sqr b) (* 4 (* a c))))) (* 2 a))))