Average Error: 35.7 → 6.3
Time: 11.5s
Precision: 64
Internal precision: 2944
\[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
⬇
\[\begin{array}{l} \mathbf{if}\;b/2 \le -8.631899346984487 \cdot 10^{+78}:\\ \;\;\;\;-2 \cdot \frac{b/2}{a}\\ \mathbf{if}\;b/2 \le 1.9124546476349704 \cdot 10^{-134}:\\ \;\;\;\;\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\\ \mathbf{if}\;b/2 \le 712713953.8918697:\\ \;\;\;\;\frac{1}{\frac{a}{c \cdot a} \cdot \left(\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b/2} \cdot \frac{-1}{2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b/2

Bits error versus c

Derivation

  1. Split input into 4 regimes.

  2. if b/2 < -8.631899346984487e+78

    1. Initial program 42.9

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

    2. Applied taylor 0

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

    3. Taylor expanded around -inf 0

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

    if -8.631899346984487e+78 < b/2 < 1.9124546476349704e-134

    1. Initial program 11.5

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

    if 1.9124546476349704e-134 < b/2 < 712713953.8918697

    1. Initial program 36.4

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

    3. Applied clear-num 36.4

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

    5. Applied flip-+ 36.6

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

    6. Applied associate-/r/ 36.6

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

    7. Applied simplify 18.5

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

    if 712713953.8918697 < b/2

    1. Initial program 58.3

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

    2. Applied taylor 14.8

      \[\leadsto \frac{\frac{-1}{2} \cdot \frac{c \cdot a}{b/2}}{a}\]

    3. Taylor expanded around inf 14.8

      \[\leadsto \frac{\color{blue}{\frac{-1}{2} \cdot \frac{c \cdot a}{b/2}}}{a}\]

    4. Applied simplify 0.0

      \[\leadsto \color{blue}{\frac{c}{b/2} \cdot \frac{-1}{2}}\]
  3. Recombined 4 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 11.5s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1064875752 1442698706 3150723005 1316518582 2592983078 3835530843)'
(FPCore (a b/2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b/2) (sqrt (- (sqr b/2) (* a c)))) a))