\[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
Test:
NMSE problem 3.2.1, positive
Bits:
128 bits
Bits error versus a
Bits error versus b/2
Bits error versus c
Time: 14.0 s
Input Error: 17.6
Output Error: 3.5
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\frac{\frac{1}{2} \cdot c}{\frac{b/2}{a}} - \left(b/2 - \left(-b/2\right)\right)}{a} & \text{when } b/2 \le -3.840384f-10 \\ \frac{\frac{a}{1} \cdot \frac{c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a} & \text{when } b/2 \le 1.7973414f+17 \\ \frac{c \cdot \frac{a}{a}}{\left(\left(-b/2\right) - b/2\right) + \frac{\frac{1}{2} \cdot a}{\frac{b/2}{c}}} & \text{otherwise} \end{cases}\)

    if b/2 < -3.840384f-10

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      17.2
    2. Using strategy rm
      17.2
    3. Applied add-exp-log to get
      \[\frac{\left(-b/2\right) + \color{red}{\sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\left(-b/2\right) + \color{blue}{e^{\log \left(\sqrt{{b/2}^2 - a \cdot c}\right)}}}{a}\]
      17.8
    4. Applied taylor to get
      \[\frac{\left(-b/2\right) + e^{\log \left(\sqrt{{b/2}^2 - a \cdot c}\right)}}{a} \leadsto \frac{\left(-b/2\right) + e^{\log \left(\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - b/2\right)}}{a}\]
      8.9
    5. Taylor expanded around -inf to get
      \[\frac{\left(-b/2\right) + e^{\log \color{red}{\left(\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - b/2\right)}}}{a} \leadsto \frac{\left(-b/2\right) + e^{\log \color{blue}{\left(\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - b/2\right)}}}{a}\]
      8.9
    6. Applied simplify to get
      \[\frac{\left(-b/2\right) + e^{\log \left(\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - b/2\right)}}{a} \leadsto \frac{\frac{\frac{1}{2} \cdot c}{\frac{b/2}{a}} - \left(b/2 - \left(-b/2\right)\right)}{a}\]
      2.3
    7. Applied final simplification

    if -3.840384f-10 < b/2 < 1.7973414f+17

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      12.3
    2. Using strategy rm
      12.3
    3. Applied flip-+ to get
      \[\frac{\color{red}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\color{blue}{\frac{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}}{a}\]
      14.6
    4. Applied simplify to get
      \[\frac{\frac{\color{red}{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\frac{\color{blue}{a \cdot c}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a}\]
      7.5
    5. Using strategy rm
      7.5
    6. Applied *-un-lft-identity to get
      \[\frac{\frac{a \cdot c}{\color{red}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}}{a} \leadsto \frac{\frac{a \cdot c}{\color{blue}{1 \cdot \left(\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}\right)}}}{a}\]
      7.5
    7. Applied times-frac to get
      \[\frac{\color{red}{\frac{a \cdot c}{1 \cdot \left(\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}\right)}}}{a} \leadsto \frac{\color{blue}{\frac{a}{1} \cdot \frac{c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}}{a}\]
      5.6

    if 1.7973414f+17 < b/2

    1. Started with
      \[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
      30.0
    2. Using strategy rm
      30.0
    3. Applied flip-+ to get
      \[\frac{\color{red}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\color{blue}{\frac{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}}{a}\]
      30.8
    4. Applied simplify to get
      \[\frac{\frac{\color{red}{{\left(-b/2\right)}^2 - {\left(\sqrt{{b/2}^2 - a \cdot c}\right)}^2}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\frac{\color{blue}{a \cdot c}}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a}\]
      17.8
    5. Using strategy rm
      17.8
    6. Applied *-un-lft-identity to get
      \[\frac{\frac{a \cdot c}{\color{red}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}}{a} \leadsto \frac{\frac{a \cdot c}{\color{blue}{1 \cdot \left(\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}\right)}}}{a}\]
      17.8
    7. Applied times-frac to get
      \[\frac{\color{red}{\frac{a \cdot c}{1 \cdot \left(\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}\right)}}}{a} \leadsto \frac{\color{blue}{\frac{a}{1} \cdot \frac{c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}}{a}\]
      17.6
    8. Applied taylor to get
      \[\frac{\frac{a}{1} \cdot \frac{c}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a} \leadsto \frac{\frac{a}{1} \cdot \frac{c}{\left(-b/2\right) - \left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}{a}\]
      7.3
    9. Taylor expanded around inf to get
      \[\frac{\frac{a}{1} \cdot \frac{c}{\left(-b/2\right) - \color{red}{\left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}}{a} \leadsto \frac{\frac{a}{1} \cdot \frac{c}{\left(-b/2\right) - \color{blue}{\left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}}{a}\]
      7.3
    10. Applied simplify to get
      \[\frac{\frac{a}{1} \cdot \frac{c}{\left(-b/2\right) - \left(b/2 - \frac{1}{2} \cdot \frac{c \cdot a}{b/2}\right)}}{a} \leadsto \frac{\frac{\frac{a}{1}}{\frac{a}{c}}}{\left(\left(-b/2\right) - b/2\right) + \frac{\frac{1}{2} \cdot c}{\frac{b/2}{a}}}\]
      4.4
    11. Applied final simplification
    12. Applied simplify to get
      \[\color{red}{\frac{\frac{\frac{a}{1}}{\frac{a}{c}}}{\left(\left(-b/2\right) - b/2\right) + \frac{\frac{1}{2} \cdot c}{\frac{b/2}{a}}}} \leadsto \color{blue}{\frac{c \cdot \frac{a}{a}}{\left(\left(-b/2\right) - b/2\right) + \frac{\frac{1}{2} \cdot a}{\frac{b/2}{c}}}}\]
      0.7
  1. Removed slow pow expressions

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))