\[\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
Test:
NMSE p42, positive
Bits:
128 bits
Bits error versus a
Bits error versus b
Bits error versus c
Time: 10.9 s
Input Error: 15.8
Output Error: 14.1
Log: ⚲
Profile: 🕒
\(\frac{\frac{4}{\frac{2}{c}}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}\)
  1. Started with
    \[\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
    15.8
  2. Using strategy rm
    15.8
  3. Applied flip-+ to get
    \[\frac{\color{red}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
    21.2
  4. Applied associate-/l/ to get
    \[\color{red}{\frac{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}} \leadsto \color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}}\]
    21.2
  5. Applied taylor to get
    \[\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)} \leadsto \frac{4 \cdot \left(c \cdot a\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}\]
    15.1
  6. Taylor expanded around inf to get
    \[\frac{\color{red}{4 \cdot \left(c \cdot a\right)}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)} \leadsto \frac{\color{blue}{4 \cdot \left(c \cdot a\right)}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}\]
    15.1
  7. Applied simplify to get
    \[\frac{4 \cdot \left(c \cdot a\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)} \leadsto \frac{\frac{\frac{4}{2} \cdot \left(c \cdot a\right)}{a}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\]
    15.0
  8. Applied final simplification
  9. Applied simplify to get
    \[\color{red}{\frac{\frac{\frac{4}{2} \cdot \left(c \cdot a\right)}{a}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}} \leadsto \color{blue}{\frac{\frac{4}{\frac{2}{c}}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}}\]
    14.1

Original test:


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