Average Error: 34.4 → 5.9
Time: 12.9s
Precision: 64
Internal precision: 2688
\[\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a}\]
⬇
\[\begin{array}{l} \mathbf{if}\;b/2 \le -9.309963021915681 \cdot 10^{+129}:\\ \;\;\;\;\frac{c}{b/2} \cdot \frac{-1}{2}\\ \mathbf{if}\;b/2 \le 5.3169981736823014 \cdot 10^{-303}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{{b/2}^2 - c \cdot a} + \left(-b/2\right)}{c}}\\ \mathbf{if}\;b/2 \le 5.782486744451723 \cdot 10^{+116}:\\ \;\;\;\;\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b/2}{a}\\ \end{array}\]

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 < -9.309963021915681e+129

    1. Initial program 61.3

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

    2. Applied taylor 14.4

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

    3. Taylor expanded around -inf 14.4

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

    4. Applied simplify 0.0

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

    if -9.309963021915681e+129 < b/2 < 5.3169981736823014e-303

    1. Initial program 34.6

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

    3. Applied flip-- 34.7

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

    4. Applied simplify 16.6

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

    6. Applied clear-num 16.8

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

    7. Applied simplify 9.4

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

    if 5.3169981736823014e-303 < b/2 < 5.782486744451723e+116

    1. Initial program 8.7

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

    if 5.782486744451723e+116 < b/2

    1. Initial program 51.2

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

    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: 12.9s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1065033997 2389885643 4100569014 2620012693 26800780 3144211646)'
(FPCore (a b/2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b/2) (sqrt (- (sqr b/2) (* a c)))) a))