Average Error: 35.0 → 6.7
Time: 34.4s
Precision: 64
Internal precision: 2944
\[\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a}\]
⬇
\[\begin{array}{l} \mathbf{if}\;b/2 \le -2.562534304360214 \cdot 10^{+50}:\\ \;\;\;\;\frac{c}{\frac{c \cdot \frac{1}{2}}{\frac{b/2}{a}} - \left(b/2 - \left(-b/2\right)\right)}\\ \mathbf{if}\;b/2 \le 3.5868674500127266 \cdot 10^{-299}:\\ \;\;\;\;\frac{\frac{a}{1} \cdot \frac{c}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}}{a}\\ \mathbf{if}\;b/2 \le 1.5428617303718363 \cdot 10^{+108}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b/2}{a}\\ \end{array}\]

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 < -2.562534304360214e+50

    1. Initial program 58.4

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

    3. Applied flip-- 58.4

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

    4. Applied simplify 32.7

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

    5. Applied taylor 16.7

      \[\leadsto \frac{\frac{a \cdot c}{\left(-b/2\right) + \left(\frac{1}{2} \cdot \frac{c \cdot a}{b/2} - b/2\right)}}{a}\]

    6. Taylor expanded around -inf 16.7

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

    7. Applied simplify 2.3

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

    if -2.562534304360214e+50 < b/2 < 3.5868674500127266e-299

    1. Initial program 29.9

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

    3. Applied flip-- 30.0

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

    4. Applied simplify 16.5

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

    6. Applied *-un-lft-identity 16.5

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

    7. Applied times-frac 13.9

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

    if 3.5868674500127266e-299 < b/2 < 1.5428617303718363e+108

    1. Initial program 8.2

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

    3. Applied clear-num 8.3

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

    if 1.5428617303718363e+108 < b/2

    1. Initial program 47.6

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

    2. Applied taylor 0

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

    3. Taylor expanded around inf 0

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

Runtime

Time bar (total: 34.4s) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(2277612311 2645429965 1090895633 2857793080 2144184008 3989768357)'
(FPCore (a b/2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b/2) (sqrt (- (sqr b/2) (* a c)))) a))