Average Error: 33.1 → 6.8
Time: 23.0s
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.5173517381925349 \cdot 10^{+153}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -2.1097631082543147 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}\\ \mathbf{elif}\;b_2 \le 8.95030690863767 \cdot 10^{+153}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b_2}{c} \cdot -2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -1.5173517381925349e+153

    1. Initial program 60.2

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

    2. Taylor expanded around -inf 2.8

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

    if -1.5173517381925349e+153 < b_2 < -2.1097631082543147e-305

    1. Initial program 8.7

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied clear-num8.9

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

    if -2.1097631082543147e-305 < b_2 < 8.95030690863767e+153

    1. Initial program 33.9

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied flip-+33.9

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

    4. Applied associate-/l/38.0

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

    5. Simplified19.3

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

    7. Applied times-frac8.0

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

    8. Simplified8.0

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

    if 8.95030690863767e+153 < b_2

    1. Initial program 62.9

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied clear-num62.9

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

    4. Taylor expanded around 0 2.5

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.5173517381925349 \cdot 10^{+153}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -2.1097631082543147 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}\\ \mathbf{elif}\;b_2 \le 8.95030690863767 \cdot 10^{+153}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b_2}{c} \cdot -2}\\ \end{array}\]

Runtime

Time bar (total: 23.0s)Debug logProfile

herbie shell --seed 2018219 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))