\[\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
Test:
NMSE p42, positive
Bits:
128 bits
Bits error versus a
Bits error versus b
Bits error versus c
Time: 27.1 s
Input Error: 34.1
Output Error: 5.8
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\frac{c}{b}}{1} - \frac{b}{a} & \text{when } b \le -2.2169129342957125 \cdot 10^{+142} \\ \frac{1}{2} \cdot \frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{a} & \text{when } b \le 1.2379375656305326 \cdot 10^{-258} \\ \frac{c}{2} \cdot \frac{4}{\left(-b\right) - \sqrt{{b}^2 - \left(c \cdot a\right) \cdot 4}} & \text{when } b \le 2.7032921376893094 \cdot 10^{+83} \\ \frac{\frac{c}{\frac{2}{4}}}{\frac{c}{b} \cdot a - b} \cdot \frac{1}{2} & \text{otherwise} \end{cases}\)

    if b < -2.2169129342957125e+142

    1. Started with
      \[\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
      57.3
    2. Using strategy rm
      57.3
    3. Applied clear-num to get
      \[\color{red}{\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}\]
      57.3
    4. Applied taylor to get
      \[\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}} \leadsto \frac{1}{\frac{2 \cdot a}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}\]
      13.7
    5. Taylor expanded around -inf to get
      \[\frac{1}{\frac{2 \cdot a}{\color{red}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}} \leadsto \frac{1}{\frac{2 \cdot a}{\color{blue}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}}\]
      13.7
    6. Applied simplify to get
      \[\frac{1}{\frac{2 \cdot a}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}} \leadsto \frac{1}{\frac{\frac{a}{1}}{a \cdot \frac{c}{b} - b}}\]
      1.3
    7. Applied final simplification
    8. Applied simplify to get
      \[\color{red}{\frac{1}{\frac{\frac{a}{1}}{a \cdot \frac{c}{b} - b}}} \leadsto \color{blue}{\frac{\frac{c}{b}}{1} - \frac{b}{a}}\]
      0.0

    if -2.2169129342957125e+142 < b < 1.2379375656305326e-258

    1. Started with
      \[\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
      9.4
    2. Using strategy rm
      9.4
    3. Applied *-un-lft-identity to get
      \[\frac{\color{red}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\color{blue}{1 \cdot \left(\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a}\]
      9.4
    4. Applied times-frac to get
      \[\color{red}{\frac{1 \cdot \left(\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}} \leadsto \color{blue}{\frac{1}{2} \cdot \frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{a}}\]
      9.4

    if 1.2379375656305326e-258 < b < 2.7032921376893094e+83

    1. Started with
      \[\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
      32.4
    2. Using strategy rm
      32.4
    3. Applied flip-+ to get
      \[\frac{\color{red}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
      32.5
    4. Applied simplify to get
      \[\frac{\frac{\color{red}{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\frac{\color{blue}{c \cdot \left(4 \cdot a\right)}}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
      16.1
    5. Using strategy rm
      16.1
    6. Applied *-un-lft-identity to get
      \[\frac{\frac{c \cdot \left(4 \cdot a\right)}{\color{red}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a} \leadsto \frac{\frac{c \cdot \left(4 \cdot a\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}}}{2 \cdot a}\]
      16.1
    7. Applied times-frac to get
      \[\frac{\color{red}{\frac{c \cdot \left(4 \cdot a\right)}{1 \cdot \left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}}}{2 \cdot a} \leadsto \frac{\color{blue}{\frac{c}{1} \cdot \frac{4 \cdot a}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
      13.7
    8. Applied times-frac to get
      \[\color{red}{\frac{\frac{c}{1} \cdot \frac{4 \cdot a}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}} \leadsto \color{blue}{\frac{\frac{c}{1}}{2} \cdot \frac{\frac{4 \cdot a}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{a}}\]
      9.0
    9. Applied simplify to get
      \[\color{red}{\frac{\frac{c}{1}}{2}} \cdot \frac{\frac{4 \cdot a}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{a} \leadsto \color{blue}{\frac{c}{2}} \cdot \frac{\frac{4 \cdot a}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{a}\]
      9.0
    10. Applied simplify to get
      \[\frac{c}{2} \cdot \color{red}{\frac{\frac{4 \cdot a}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{a}} \leadsto \frac{c}{2} \cdot \color{blue}{\frac{4}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}}\]
      7.7
    11. Applied simplify to get
      \[\frac{c}{2} \cdot \frac{4}{\color{red}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}} \leadsto \frac{c}{2} \cdot \frac{4}{\color{blue}{\left(-b\right) - \sqrt{{b}^2 - \left(c \cdot a\right) \cdot 4}}}\]
      7.7

    if 2.7032921376893094e+83 < b

    1. Started with
      \[\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
      58.9
    2. Using strategy rm
      58.9
    3. Applied flip-+ to get
      \[\frac{\color{red}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
      58.9
    4. Applied simplify to get
      \[\frac{\frac{\color{red}{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\frac{\color{blue}{c \cdot \left(4 \cdot a\right)}}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
      33.5
    5. Applied taylor to get
      \[\frac{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\frac{c \cdot \left(4 \cdot a\right)}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a}\]
      16.2
    6. Taylor expanded around inf to get
      \[\frac{\frac{c \cdot \left(4 \cdot a\right)}{\color{red}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}}{2 \cdot a} \leadsto \frac{\frac{c \cdot \left(4 \cdot a\right)}{\color{blue}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}}{2 \cdot a}\]
      16.2
    7. Applied simplify to get
      \[\color{red}{\frac{\frac{c \cdot \left(4 \cdot a\right)}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a}} \leadsto \color{blue}{\frac{\frac{c}{\frac{2}{4}}}{\frac{c}{b} \cdot a - b} \cdot \frac{1}{2}}\]
      1.7
  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))))))