\[\frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
Test:
The quadratic formula (r1)
Bits:
128 bits
Bits error versus a
Bits error versus b
Bits error versus c
Time: 12.4 s
Input Error: 16.0
Output Error: 13.7
Log: ⚲
Profile: 🕒
\(\frac{1}{\left(\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{\frac{2}{4}}{c}}\)
  1. Started with
    \[\frac{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
    16.0
  2. Using strategy rm
    16.0
  3. Applied flip-+ to get
    \[\frac{\color{red}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]
    21.0
  4. Applied simplify to get
    \[\frac{\frac{\color{red}{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \leadsto \frac{\frac{\color{blue}{\left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
    14.7
  5. Using strategy rm
    14.7
  6. Applied clear-num to get
    \[\color{red}{\frac{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}} \leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}}}\]
    14.7
  7. Applied simplify to get
    \[\frac{1}{\color{red}{\frac{2 \cdot a}{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}}} \leadsto \frac{1}{\color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}\right) \cdot \frac{\frac{2}{4}}{c}}}\]
    13.8
  8. Applied simplify to get
    \[\frac{1}{\color{red}{\left(\left(-b\right) - \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}\right)} \cdot \frac{\frac{2}{4}}{c}} \leadsto \frac{1}{\color{blue}{\left(\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)} \cdot \frac{\frac{2}{4}}{c}}\]
    13.7

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))))))