Average Error: 34.5 → 7.4
Time: 24.4s
Precision: 64
Internal precision: 2944
\[\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a}\]
⬇
\[\begin{array}{l} \mathbf{if}\;b/2 \le -7.275666222599798 \cdot 10^{+72}:\\ \;\;\;\;\frac{c}{b/2} \cdot \frac{-1}{2}\\ \mathbf{if}\;b/2 \le -5.012107324121991 \cdot 10^{-257}:\\ \;\;\;\;\frac{c \cdot a}{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}} \cdot \frac{1}{a}\\ \mathbf{if}\;b/2 \le 2.6854465913894807 \cdot 10^{+99}:\\ \;\;\;\;\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b/2}{a}\\ \end{array}\]

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 < -7.275666222599798e+72

    1. Initial program 58.4

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

    2. Applied taylor 15.1

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

    3. Taylor expanded around -inf 15.1

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

    4. Applied simplify 0.0

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

    if -7.275666222599798e+72 < b/2 < -5.012107324121991e-257

    1. Initial program 32.2

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

    3. Applied div-inv 32.3

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

    5. Applied flip-- 32.4

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

    6. Applied simplify 17.0

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

    if -5.012107324121991e-257 < b/2 < 2.6854465913894807e+99

    1. Initial program 9.9

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

    if 2.6854465913894807e+99 < b/2

    1. Initial program 46.3

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

    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: 24.4s) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(644180380 3784176976 401987740 22459203 1940947670 3323606534)'
(FPCore (a b/2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b/2) (sqrt (- (sqr b/2) (* a c)))) a))