\[\frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
Test:
The quadratic formula (r1)
Bits:
128 bits
Bits error versus a
Bits error versus b
Bits error versus c
Time: 32.1 s
Input Error: 34.8
Output Error: 6.1
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\frac{2 \cdot c}{\frac{b}{a}} - \left(b - \left(-b\right)\right)}{a \cdot 2} & \text{when } b \le -2.326207163052239 \cdot 10^{+115} \\ \frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \le -1.9116038778178358 \cdot 10^{-274} \\ \frac{1}{2} \cdot \frac{4 \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(c \cdot a\right) \cdot 4}} & \text{when } b \le 4.449432714488087 \cdot 10^{+57} \\ \frac{\frac{4}{2} \cdot c}{(\left(a \cdot 2\right) * \left(\frac{c}{b}\right) + \left(-b\right))_* - b} & \text{otherwise} \end{cases}\)

    if b < -2.326207163052239e+115

    1. Started with
      \[\frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
      48.9
    2. Using strategy rm
      48.9
    3. Applied add-cube-cbrt to get
      \[\frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{\color{red}{2 \cdot a}} \leadsto \frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{\color{blue}{{\left(\sqrt[3]{2 \cdot a}\right)}^3}}\]
      49.2
    4. Applied add-cube-cbrt to get
      \[\frac{\color{red}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{{\left(\sqrt[3]{2 \cdot a}\right)}^3} \leadsto \frac{\color{blue}{{\left(\sqrt[3]{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}\right)}^3}}{{\left(\sqrt[3]{2 \cdot a}\right)}^3}\]
      49.2
    5. Applied cube-undiv to get
      \[\color{red}{\frac{{\left(\sqrt[3]{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}\right)}^3}{{\left(\sqrt[3]{2 \cdot a}\right)}^3}} \leadsto \color{blue}{{\left(\frac{\sqrt[3]{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{\sqrt[3]{2 \cdot a}}\right)}^3}\]
      49.2
    6. Applied taylor to get
      \[{\left(\frac{\sqrt[3]{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{\sqrt[3]{2 \cdot a}}\right)}^3 \leadsto {\left(\frac{\sqrt[3]{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)}}{\sqrt[3]{2 \cdot a}}\right)}^3\]
      13.1
    7. Taylor expanded around -inf to get
      \[{\left(\frac{\sqrt[3]{\left(-b\right) + \color{red}{\left(2 \cdot \frac{c \cdot a}{b} - b\right)}}}{\sqrt[3]{2 \cdot a}}\right)}^3 \leadsto {\left(\frac{\sqrt[3]{\left(-b\right) + \color{blue}{\left(2 \cdot \frac{c \cdot a}{b} - b\right)}}}{\sqrt[3]{2 \cdot a}}\right)}^3\]
      13.1
    8. Applied simplify to get
      \[{\left(\frac{\sqrt[3]{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)}}{\sqrt[3]{2 \cdot a}}\right)}^3 \leadsto \frac{\frac{2 \cdot c}{\frac{b}{a}} - \left(b - \left(-b\right)\right)}{a \cdot 2}\]
      1.4
    9. Applied final simplification

    if -2.326207163052239e+115 < b < -1.9116038778178358e-274

    1. Started with
      \[\frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
      9.0

    if -1.9116038778178358e-274 < b < 4.449432714488087e+57

    1. Started with
      \[\frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
      29.1
    2. Using strategy rm
      29.1
    3. Applied flip-+ to get
      \[\frac{\color{red}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]
      29.2
    4. Applied simplify to get
      \[\frac{\frac{\color{red}{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \leadsto \frac{\frac{\color{blue}{\left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
      17.0
    5. Using strategy rm
      17.0
    6. Applied *-un-lft-identity to get
      \[\frac{\color{red}{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a} \leadsto \frac{\color{blue}{1 \cdot \frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]
      17.0
    7. Applied times-frac to get
      \[\color{red}{\frac{1 \cdot \frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}} \leadsto \color{blue}{\frac{1}{2} \cdot \frac{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{a}}\]
      17.0
    8. Applied simplify to get
      \[\frac{1}{2} \cdot \color{red}{\frac{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{a}} \leadsto \frac{1}{2} \cdot \color{blue}{\frac{4 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}}\]
      10.2
    9. Applied simplify to get
      \[\frac{1}{2} \cdot \frac{4 \cdot c}{\color{red}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}} \leadsto \frac{1}{2} \cdot \frac{4 \cdot c}{\color{blue}{\left(-b\right) - \sqrt{{b}^2 - \left(c \cdot a\right) \cdot 4}}}\]
      10.2

    if 4.449432714488087e+57 < b

    1. Started with
      \[\frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
      58.3
    2. Using strategy rm
      58.3
    3. Applied flip-+ to get
      \[\frac{\color{red}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]
      58.3
    4. Applied simplify to get
      \[\frac{\frac{\color{red}{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \leadsto \frac{\frac{\color{blue}{\left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
      32.4
    5. Applied taylor to get
      \[\frac{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \leadsto \frac{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \left(b - 2 \cdot \frac{c \cdot a}{b}\right)}}{2 \cdot a}\]
      15.3
    6. Taylor expanded around inf to get
      \[\frac{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \color{red}{\left(b - 2 \cdot \frac{c \cdot a}{b}\right)}}}{2 \cdot a} \leadsto \frac{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \color{blue}{\left(b - 2 \cdot \frac{c \cdot a}{b}\right)}}}{2 \cdot a}\]
      15.3
    7. Applied simplify to get
      \[\color{red}{\frac{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \left(b - 2 \cdot \frac{c \cdot a}{b}\right)}}{2 \cdot a}} \leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(4 \cdot c\right)}{(\left(a \cdot 2\right) * \left(\frac{c}{b}\right) + \left(-b\right))_* - b}}\]
      1.6
    8. Applied simplify to get
      \[\frac{\color{red}{\frac{1}{2} \cdot \left(4 \cdot c\right)}}{(\left(a \cdot 2\right) * \left(\frac{c}{b}\right) + \left(-b\right))_* - b} \leadsto \frac{\color{blue}{\frac{4}{2} \cdot c}}{(\left(a \cdot 2\right) * \left(\frac{c}{b}\right) + \left(-b\right))_* - b}\]
      1.6
  1. Removed slow pow expressions

Original test:


(lambda ((a default) (b default) (c default))
  #:name "The quadratic formula (r1)"
  (/ (+ (- 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))))))