Average Error: 39.1 → 6.8
Time: 32.8s
Precision: 64
Internal precision: 3200
\[\frac{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}{a}\]
⬇
\[\begin{array}{l} \mathbf{if}\;b/2 \le -5.774133988795908 \cdot 10^{-117}:\\ \;\;\;\;\frac{c}{\frac{a \cdot \frac{1}{2}}{\frac{b/2}{c}} - \left(b/2 - \left(-b/2\right)\right)}\\ \mathbf{if}\;b/2 \le 1.0612379322957353 \cdot 10^{+83}:\\ \;\;\;\;\frac{-b/2}{a} - \frac{\sqrt{{b/2}^2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b/2}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b/2

Bits error versus c

Derivation

  1. Split input into 3 regimes.

  2. if b/2 < -5.774133988795908e-117

    1. Initial program 59.0

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

    3. Applied flip-- 59.0

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

    4. Applied simplify 36.7

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

    6. Applied add-cube-cbrt 36.8

      \[\leadsto \frac{\frac{a \cdot c}{\left(-b/2\right) + \color{blue}{{\left(\sqrt[3]{\sqrt{{b/2}^2 - a \cdot c}}\right)}^3}}}{a}\]

    7. Applied taylor 17.1

      \[\leadsto \frac{\frac{a \cdot c}{\left(-b/2\right) + {\left(\sqrt[3]{\frac{1}{2} \cdot \frac{a \cdot c}{b/2} - b/2}\right)}^3}}{a}\]

    8. Taylor expanded around -inf 17.1

      \[\leadsto \frac{\frac{a \cdot c}{\left(-b/2\right) + {\left(\sqrt[3]{\color{blue}{\frac{1}{2} \cdot \frac{a \cdot c}{b/2} - b/2}}\right)}^3}}{a}\]

    9. Applied simplify 5.1

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

    if -5.774133988795908e-117 < b/2 < 1.0612379322957353e+83

    1. Initial program 11.9

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

    3. Applied div-sub 11.9

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

    if 1.0612379322957353e+83 < b/2

    1. Initial program 45.4

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

    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 3 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 32.8s) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(3283856077 3183919125 3399458751 396155847 3688194147 1862413033)'
(FPCore (a b/2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b/2) (sqrt (- (sqr b/2) (* a c)))) a))