Average Error: 32.9 → 6.2
Time: 29.6s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.628278339169478 \cdot 10^{+149}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.5161853332160458 \cdot 10^{-263}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) + \sqrt{{b_2}^{2} - c \cdot a}}\\ \mathbf{elif}\;b_2 \le 2.0371016472469965 \cdot 10^{+152}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{{b_2}^{2} - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -3.628278339169478e+149

    1. Initial program 62.6

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around inf 62.6

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]
    3. Using strategy rm

    4. Applied div-inv62.6

      \[\leadsto \color{blue}{\left(\left(-b_2\right) - \sqrt{{b_2}^{2} - a \cdot c}\right) \cdot \frac{1}{a}}\]

    5. Taylor expanded around -inf 1.5

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]

    if -3.628278339169478e+149 < b_2 < 1.5161853332160458e-263

    1. Initial program 32.6

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around inf 32.6

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]
    3. Using strategy rm

    4. Applied div-inv32.7

      \[\leadsto \color{blue}{\left(\left(-b_2\right) - \sqrt{{b_2}^{2} - a \cdot c}\right) \cdot \frac{1}{a}}\]
    5. Using strategy rm

    6. Applied flip--32.8

      \[\leadsto \color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{{b_2}^{2} - a \cdot c} \cdot \sqrt{{b_2}^{2} - a \cdot c}}{\left(-b_2\right) + \sqrt{{b_2}^{2} - a \cdot c}}} \cdot \frac{1}{a}\]

    7. Applied associate-*l/32.8

      \[\leadsto \color{blue}{\frac{\left(\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{{b_2}^{2} - a \cdot c} \cdot \sqrt{{b_2}^{2} - a \cdot c}\right) \cdot \frac{1}{a}}{\left(-b_2\right) + \sqrt{{b_2}^{2} - a \cdot c}}}\]

    8. Simplified13.4

      \[\leadsto \frac{\color{blue}{\frac{0 + a \cdot c}{a}}}{\left(-b_2\right) + \sqrt{{b_2}^{2} - a \cdot c}}\]

    9. Taylor expanded around -inf 7.9

      \[\leadsto \frac{\color{blue}{c}}{\left(-b_2\right) + \sqrt{{b_2}^{2} - a \cdot c}}\]

    if 1.5161853332160458e-263 < b_2 < 2.0371016472469965e+152

    1. Initial program 8.0

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around inf 8.0

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]

    if 2.0371016472469965e+152 < b_2

    1. Initial program 59.8

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around inf 59.8

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]
    3. Using strategy rm

    4. Applied div-inv59.8

      \[\leadsto \color{blue}{\left(\left(-b_2\right) - \sqrt{{b_2}^{2} - a \cdot c}\right) \cdot \frac{1}{a}}\]

    5. Taylor expanded around inf 2.3

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.628278339169478 \cdot 10^{+149}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.5161853332160458 \cdot 10^{-263}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) + \sqrt{{b_2}^{2} - c \cdot a}}\\ \mathbf{elif}\;b_2 \le 2.0371016472469965 \cdot 10^{+152}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{{b_2}^{2} - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]

Runtime

Time bar (total: 29.6s)Debug logProfile

herbie shell --seed 2018296 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))