Average Error: 33.3 → 12.3
Time: 30.1s
Precision: 64
Internal Precision: 3136
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.263114766361561 \cdot 10^{+105}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.8994027553436794 \cdot 10^{-111}:\\ \;\;\;\;(\left(\frac{1}{a}\right) \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right) + \left(\frac{-b_2}{a}\right))_*\\ \mathbf{elif}\;b_2 \le 5.856983612190039 \cdot 10^{+123}:\\ \;\;\;\;\frac{-a \cdot c}{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot c}{b_2} \cdot \frac{-1}{2}}{a}\\ \end{array}\]

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 < -1.263114766361561e+105

    1. Initial program 45.7

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

    2. Initial simplification45.7

      \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\]

    3. Taylor expanded around 0 45.7

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

    4. Taylor expanded around -inf 3.5

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

    if -1.263114766361561e+105 < b_2 < 1.8994027553436794e-111

    1. Initial program 12.1

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

    2. Initial simplification12.1

      \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\]

    3. Taylor expanded around 0 12.1

      \[\leadsto \frac{\sqrt{\color{blue}{{b_2}^{2} - a \cdot c}} - b_2}{a}\]
    4. Using strategy rm

    5. Applied div-sub12.2

      \[\leadsto \color{blue}{\frac{\sqrt{{b_2}^{2} - a \cdot c}}{a} - \frac{b_2}{a}}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt12.4

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

    8. Applied div-inv12.4

      \[\leadsto \color{blue}{\sqrt{{b_2}^{2} - a \cdot c} \cdot \frac{1}{a}} - \left(\sqrt[3]{\frac{b_2}{a}} \cdot \sqrt[3]{\frac{b_2}{a}}\right) \cdot \sqrt[3]{\frac{b_2}{a}}\]

    9. Applied prod-diff12.4

      \[\leadsto \color{blue}{(\left(\sqrt{{b_2}^{2} - a \cdot c}\right) \cdot \left(\frac{1}{a}\right) + \left(-\sqrt[3]{\frac{b_2}{a}} \cdot \left(\sqrt[3]{\frac{b_2}{a}} \cdot \sqrt[3]{\frac{b_2}{a}}\right)\right))_* + (\left(-\sqrt[3]{\frac{b_2}{a}}\right) \cdot \left(\sqrt[3]{\frac{b_2}{a}} \cdot \sqrt[3]{\frac{b_2}{a}}\right) + \left(\sqrt[3]{\frac{b_2}{a}} \cdot \left(\sqrt[3]{\frac{b_2}{a}} \cdot \sqrt[3]{\frac{b_2}{a}}\right)\right))_*}\]

    10. Simplified12.2

      \[\leadsto \color{blue}{(\left(\frac{1}{a}\right) \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right) + \left(-\frac{b_2}{a}\right))_*} + (\left(-\sqrt[3]{\frac{b_2}{a}}\right) \cdot \left(\sqrt[3]{\frac{b_2}{a}} \cdot \sqrt[3]{\frac{b_2}{a}}\right) + \left(\sqrt[3]{\frac{b_2}{a}} \cdot \left(\sqrt[3]{\frac{b_2}{a}} \cdot \sqrt[3]{\frac{b_2}{a}}\right)\right))_*\]

    11. Simplified12.2

      \[\leadsto (\left(\frac{1}{a}\right) \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right) + \left(-\frac{b_2}{a}\right))_* + \color{blue}{0}\]

    if 1.8994027553436794e-111 < b_2 < 5.856983612190039e+123

    1. Initial program 40.8

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

    2. Initial simplification40.8

      \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\]
    3. Using strategy rm

    4. Applied flip--40.8

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c} - b_2 \cdot b_2}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}}}{a}\]

    5. Applied associate-/l/43.4

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

    6. Simplified17.4

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

    if 5.856983612190039e+123 < b_2

    1. Initial program 60.3

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

    2. Initial simplification60.3

      \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\]

    3. Taylor expanded around 0 60.3

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

    4. Taylor expanded around inf 13.7

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

  4. Final simplification12.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.263114766361561 \cdot 10^{+105}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.8994027553436794 \cdot 10^{-111}:\\ \;\;\;\;(\left(\frac{1}{a}\right) \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right) + \left(\frac{-b_2}{a}\right))_*\\ \mathbf{elif}\;b_2 \le 5.856983612190039 \cdot 10^{+123}:\\ \;\;\;\;\frac{-a \cdot c}{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot c}{b_2} \cdot \frac{-1}{2}}{a}\\ \end{array}\]

Runtime

Time bar (total: 30.1s)Debug logProfile

herbie shell --seed 2018251 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))