\[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
Test:
The quadratic formula (r2)
Bits:
128 bits
Bits error versus a
Bits error versus b
Bits error versus c
Time: 9.1 s
Input Error: 15.7
Output Error: 15.7
Log: ⚲
Profile: 🕒
\(\frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\)
  1. Started with
    \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
    15.7
  2. Using strategy rm
    15.7
  3. Applied pow1 to get
    \[\frac{\color{red}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\color{blue}{{\left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^{1}}}{2 \cdot a}\]
    15.7
  4. Applied taylor to get
    \[\frac{{\left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^{1}}{2 \cdot a} \leadsto \frac{{\left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(c \cdot a\right)}\right)}^{1}}{2 \cdot a}\]
    15.7
  5. Taylor expanded around 0 to get
    \[\frac{{\left(\left(-b\right) - \sqrt{\color{red}{{b}^2 - 4 \cdot \left(c \cdot a\right)}}\right)}^{1}}{2 \cdot a} \leadsto \frac{{\left(\left(-b\right) - \sqrt{\color{blue}{{b}^2 - 4 \cdot \left(c \cdot a\right)}}\right)}^{1}}{2 \cdot a}\]
    15.7
  6. Applied simplify to get
    \[\color{red}{\frac{{\left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(c \cdot a\right)}\right)}^{1}}{2 \cdot a}} \leadsto \color{blue}{\frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}}\]
    15.7

Original test:


(lambda ((a default) (b default) (c default))
  #:name "The quadratic formula (r2)"
  (/ (- (- b) (sqrt (- (sqr b) (* 4 (* a c))))) (* 2 a))
  #:target
  (if (< b 0) (/ c (* a (/ (+ (- b) (sqrt (- (sqr b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (sqr b) (* 4 (* a c))))) (* 2 a))))