\[\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
Test:
NMSE p42, positive
Bits:
128 bits
Bits error versus a
Bits error versus b
Bits error versus c
Time: 19.2 s
Input Error: 16.0
Output Error: 2.6
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{c}{b} - \frac{b}{a} & \text{when } b \le -1.2459677f+19 \\ \frac{\left(-\sqrt{{b}^2 - a \cdot \left(4 \cdot c\right)}\right) + b}{-2 \cdot a} & \text{when } b \le -7.39763f-38 \\ \frac{c}{2} \cdot \frac{4}{\left(-b\right) - \sqrt{{b}^2 - \left(c \cdot a\right) \cdot 4}} & \text{when } b \le 2.441097f+08 \\ \frac{c}{-b} & \text{otherwise} \end{cases}\)

    if b < -1.2459677f+19

    1. Started with
      \[\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
      28.6
    2. Applied taylor to get
      \[\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \leadsto \frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{2 \cdot a}\]
      6.1
    3. Taylor expanded around -inf to get
      \[\frac{\color{red}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a} \leadsto \frac{\color{blue}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a}\]
      6.1
    4. Applied simplify to get
      \[\color{red}{\frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{2 \cdot a}} \leadsto \color{blue}{\frac{\frac{c}{b}}{1} - \frac{b}{a}}\]
      0.0
    5. Applied simplify to get
      \[\color{red}{\frac{\frac{c}{b}}{1}} - \frac{b}{a} \leadsto \color{blue}{\frac{c}{b}} - \frac{b}{a}\]
      0.0

    if -1.2459677f+19 < b < -7.39763f-38

    1. Started with
      \[\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
      4.0
    2. Using strategy rm
      4.0
    3. Applied frac-2neg to get
      \[\color{red}{\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \leadsto \color{blue}{\frac{-\left(\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}{-2 \cdot a}}\]
      4.0
    4. Applied simplify to get
      \[\frac{\color{red}{-\left(\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}}{-2 \cdot a} \leadsto \frac{\color{blue}{\left(-\sqrt{{b}^2 - a \cdot \left(4 \cdot c\right)}\right) + b}}{-2 \cdot a}\]
      4.0

    if -7.39763f-38 < b < 2.441097f+08

    1. Started with
      \[\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
      13.1
    2. Using strategy rm
      13.1
    3. Applied flip-+ to get
      \[\frac{\color{red}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
      13.6
    4. Applied simplify to get
      \[\frac{\frac{\color{red}{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\frac{\color{blue}{c \cdot \left(4 \cdot a\right)}}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
      7.4
    5. Using strategy rm
      7.4
    6. Applied *-un-lft-identity to get
      \[\frac{\frac{c \cdot \left(4 \cdot a\right)}{\color{red}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a} \leadsto \frac{\frac{c \cdot \left(4 \cdot a\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}}}{2 \cdot a}\]
      7.4
    7. Applied times-frac to get
      \[\frac{\color{red}{\frac{c \cdot \left(4 \cdot a\right)}{1 \cdot \left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}}}{2 \cdot a} \leadsto \frac{\color{blue}{\frac{c}{1} \cdot \frac{4 \cdot a}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
      4.6
    8. Applied times-frac to get
      \[\color{red}{\frac{\frac{c}{1} \cdot \frac{4 \cdot a}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}} \leadsto \color{blue}{\frac{\frac{c}{1}}{2} \cdot \frac{\frac{4 \cdot a}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{a}}\]
      4.6
    9. Applied simplify to get
      \[\color{red}{\frac{\frac{c}{1}}{2}} \cdot \frac{\frac{4 \cdot a}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{a} \leadsto \color{blue}{\frac{c}{2}} \cdot \frac{\frac{4 \cdot a}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{a}\]
      4.6
    10. Applied simplify to get
      \[\frac{c}{2} \cdot \color{red}{\frac{\frac{4 \cdot a}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{a}} \leadsto \frac{c}{2} \cdot \color{blue}{\frac{4}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}}\]
      4.6
    11. Applied simplify to get
      \[\frac{c}{2} \cdot \frac{4}{\color{red}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}} \leadsto \frac{c}{2} \cdot \frac{4}{\color{blue}{\left(-b\right) - \sqrt{{b}^2 - \left(c \cdot a\right) \cdot 4}}}\]
      4.5

    if 2.441097f+08 < b

    1. Started with
      \[\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
      28.3
    2. Using strategy rm
      28.3
    3. Applied flip-+ to get
      \[\frac{\color{red}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
      30.2
    4. Applied simplify to get
      \[\frac{\frac{\color{red}{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\frac{\color{blue}{c \cdot \left(4 \cdot a\right)}}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
      15.4
    5. Using strategy rm
      15.4
    6. Applied clear-num to get
      \[\color{red}{\frac{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}} \leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}}\]
      15.4
    7. Applied simplify to get
      \[\frac{1}{\color{red}{\frac{2 \cdot a}{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}} \leadsto \frac{1}{\color{blue}{\frac{2}{4 \cdot c} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}\]
      15.0
    8. Applied taylor to get
      \[\frac{1}{\frac{2}{4 \cdot c} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)} \leadsto \frac{1}{-1 \cdot \frac{b}{c}}\]
      1.1
    9. Taylor expanded around 0 to get
      \[\frac{1}{\color{red}{-1 \cdot \frac{b}{c}}} \leadsto \frac{1}{\color{blue}{-1 \cdot \frac{b}{c}}}\]
      1.1
    10. Applied simplify to get
      \[\frac{1}{-1 \cdot \frac{b}{c}} \leadsto \frac{\frac{1}{-1}}{\frac{b}{c}}\]
      1.1
    11. Applied final simplification
    12. Applied simplify to get
      \[\color{red}{\frac{\frac{1}{-1}}{\frac{b}{c}}} \leadsto \color{blue}{\frac{c}{-b}}\]
      0
  1. Removed slow pow expressions

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))))))