\[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
Test:
NMSE p42, negative
Bits:
128 bits
Bits error versus a
Bits error versus b
Bits error versus c
Time: 10.5 s
Input Error: 16.0
Output Error: 16.7
Log: ⚲
Profile: 🕒
\((\left(\sqrt{{b}^2 - \left(4 \cdot c\right) \cdot a}\right) * \left(\frac{\frac{-1}{2}}{a}\right) + \left(\frac{\frac{-1}{2}}{\frac{a}{b}}\right))_*\)
  1. Started with
    \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
    16.0
  2. Using strategy rm
    16.0
  3. Applied add-cube-cbrt to get
    \[\color{red}{\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\right)}^3}\]
    16.3
  4. Applied taylor to get
    \[{\left(\sqrt[3]{\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\right)}^3 \leadsto {\left(\sqrt[3]{\frac{-1}{2} \cdot \left(\left(\frac{1}{b} + \sqrt{\frac{1}{{b}^2} - 4 \cdot \frac{1}{c \cdot a}}\right) \cdot a\right)}\right)}^3\]
    29.8
  5. Taylor expanded around inf to get
    \[{\color{red}{\left(\sqrt[3]{\frac{-1}{2} \cdot \left(\left(\frac{1}{b} + \sqrt{\frac{1}{{b}^2} - 4 \cdot \frac{1}{c \cdot a}}\right) \cdot a\right)}\right)}}^3 \leadsto {\color{blue}{\left(\sqrt[3]{\frac{-1}{2} \cdot \left(\left(\frac{1}{b} + \sqrt{\frac{1}{{b}^2} - 4 \cdot \frac{1}{c \cdot a}}\right) \cdot a\right)}\right)}}^3\]
    29.8
  6. Applied simplify to get
    \[\color{red}{{\left(\sqrt[3]{\frac{-1}{2} \cdot \left(\left(\frac{1}{b} + \sqrt{\frac{1}{{b}^2} - 4 \cdot \frac{1}{c \cdot a}}\right) \cdot a\right)}\right)}^3} \leadsto \color{blue}{(\left(\sqrt{\frac{1}{b \cdot b} - \frac{\frac{4}{a}}{c}}\right) * \left(\frac{-1}{2} \cdot a\right) + \left(\frac{a}{b} \cdot \frac{-1}{2}\right))_*}\]
    29.9
  7. Applied taylor to get
    \[(\left(\sqrt{\frac{1}{b \cdot b} - \frac{\frac{4}{a}}{c}}\right) * \left(\frac{-1}{2} \cdot a\right) + \left(\frac{a}{b} \cdot \frac{-1}{2}\right))_* \leadsto (\left(\sqrt{{b}^2 - 4 \cdot \left(c \cdot a\right)}\right) * \left(\frac{\frac{-1}{2}}{a}\right) + \left(\frac{-1}{2} \cdot \frac{b}{a}\right))_*\]
    16.6
  8. Taylor expanded around inf to get
    \[\color{red}{(\left(\sqrt{{b}^2 - 4 \cdot \left(c \cdot a\right)}\right) * \left(\frac{\frac{-1}{2}}{a}\right) + \left(\frac{-1}{2} \cdot \frac{b}{a}\right))_*} \leadsto \color{blue}{(\left(\sqrt{{b}^2 - 4 \cdot \left(c \cdot a\right)}\right) * \left(\frac{\frac{-1}{2}}{a}\right) + \left(\frac{-1}{2} \cdot \frac{b}{a}\right))_*}\]
    16.6
  9. Applied simplify to get
    \[(\left(\sqrt{{b}^2 - 4 \cdot \left(c \cdot a\right)}\right) * \left(\frac{\frac{-1}{2}}{a}\right) + \left(\frac{-1}{2} \cdot \frac{b}{a}\right))_* \leadsto (\left(\sqrt{{b}^2 - \left(4 \cdot c\right) \cdot a}\right) * \left(\frac{\frac{-1}{2}}{a}\right) + \left(\frac{\frac{-1}{2}}{\frac{a}{b}}\right))_*\]
    16.7
  10. Applied final simplification

Original test:


(lambda ((a default) (b default) (c default))
  #:name "NMSE p42, negative"
  (/ (- (- 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))))