\[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
Test:
The quadratic formula (r2)
Bits:
128 bits
Bits error versus a
Bits error versus b
Bits error versus c
Time: 20.6 s
Input Error: 17.1
Output Error: 3.4
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{c}{b} \cdot \frac{-2}{2} & \text{when } b \le -0.15658416f0 \\ \frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} & \text{when } b \le 9.210374f+08 \\ \frac{c}{b} - \frac{b}{a} & \text{otherwise} \end{cases}\)

    if b < -0.15658416f0

    1. Started with
      \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
      28.1
    2. Applied taylor to get
      \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \leadsto \frac{-2 \cdot \frac{c \cdot a}{b}}{2 \cdot a}\]
      7.2
    3. Taylor expanded around -inf to get
      \[\frac{\color{red}{-2 \cdot \frac{c \cdot a}{b}}}{2 \cdot a} \leadsto \frac{\color{blue}{-2 \cdot \frac{c \cdot a}{b}}}{2 \cdot a}\]
      7.2
    4. Applied simplify to get
      \[\color{red}{\frac{-2 \cdot \frac{c \cdot a}{b}}{2 \cdot a}} \leadsto \color{blue}{\frac{c}{b} \cdot \frac{-2}{2}}\]
      0

    if -0.15658416f0 < b < 9.210374f+08

    1. Started with
      \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
      7.3

    if 9.210374f+08 < b

    1. Started with
      \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
      21.0
    2. Applied taylor to get
      \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \leadsto \frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{2 \cdot a}\]
      5.9
    3. Taylor expanded around inf to get
      \[\frac{\color{red}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a} \leadsto \frac{\color{blue}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a}\]
      5.9
    4. Applied simplify to get
      \[\color{red}{\frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{2 \cdot a}} \leadsto \color{blue}{\frac{\frac{c}{b}}{1} - \frac{b}{a}}\]
      0.0
    5. Applied simplify to get
      \[\color{red}{\frac{\frac{c}{b}}{1}} - \frac{b}{a} \leadsto \color{blue}{\frac{c}{b}} - \frac{b}{a}\]
      0.0
  1. Removed slow pow expressions

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))))