Average Error: 34.7 → 5.1
Time: 22.2s
Precision: 64
Internal precision: 2944
\[\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
⬇
\[\begin{array}{l} \mathbf{if}\;b \le -8.631899346984487 \cdot 10^{+78}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{if}\;b \le 2.0911561750607134 \cdot 10^{-260}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{if}\;b \le 3.356118893853414 \cdot 10^{+78}:\\ \;\;\;\;\frac{1}{\frac{\frac{2}{1}}{c \cdot 4} \cdot \left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-2}{2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.7
Comparison21.8
Herbie5.1
\[ \begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array} \]

Derivation

  1. Split input into 4 regimes.

  2. if b < -8.631899346984487e+78

    1. Initial program 42.9

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

    2. Applied taylor 0

      \[\leadsto -1 \cdot \frac{b}{a}\]

    3. Taylor expanded around -inf 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}}\]

    4. Applied simplify 0

      \[\leadsto \color{blue}{\frac{-b}{a}}\]

    if -8.631899346984487e+78 < b < 2.0911561750607134e-260

    1. Initial program 9.5

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

    if 2.0911561750607134e-260 < b < 3.356118893853414e+78

    1. Initial program 33.1

      \[\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied clear-num 33.1

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}\]
    4. Using strategy rm

    5. Applied flip-+ 33.2

      \[\leadsto \frac{1}{\frac{2 \cdot a}{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}}\]

    6. Applied associate-/r/ 33.3

      \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot a}{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2} \cdot \left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}}\]

    7. Applied simplify 8.9

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{a + a}{a}}{c \cdot 4}} \cdot \left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}\]

    8. Applied simplify 8.9

      \[\leadsto \frac{1}{\frac{\color{blue}{\frac{2}{1}}}{c \cdot 4} \cdot \left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}\]

    if 3.356118893853414e+78 < b

    1. Initial program 58.7

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

    2. Applied taylor 14.7

      \[\leadsto \frac{-2 \cdot \frac{c \cdot a}{b}}{2 \cdot a}\]

    3. Taylor expanded around inf 14.7

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c \cdot a}{b}}}{2 \cdot a}\]

    4. Applied simplify 0

      \[\leadsto \color{blue}{\frac{c}{b} \cdot \frac{-2}{2}}\]
  3. Recombined 4 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 22.2s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1064875752 1442698706 3150723005 1316518582 2592983078 3835530843)'
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :target
  (if (< b 0) (/ (+ (- b) (sqrt (- (sqr b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (sqr b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (sqr b) (* 4 (* a c))))) (* 2 a)))