Average Error: 33.4 → 8.9
Time: 1.6m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{-\frac{1}{2}}{b_2} \le -1660651.8925853842:\\ \;\;\;\;\frac{\frac{c \cdot a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \mathbf{if}\;\frac{-\frac{1}{2}}{b_2} \le 5.7722669403334366 \cdot 10^{-300}:\\ \;\;\;\;\frac{c}{\frac{\frac{1}{2} \cdot a}{\frac{b_2}{c}} - 2 \cdot b_2}\\ \mathbf{if}\;\frac{-\frac{1}{2}}{b_2} \le 3.4884196960522 \cdot 10^{-155}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \end{array}\]

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 (/ (- 1/2) b_2) < -1660651.8925853842

    1. Initial program 25.2

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

    3. Applied flip-+25.4

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

    4. Applied simplify17.6

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

    if -1660651.8925853842 < (/ (- 1/2) b_2) < 5.7722669403334366e-300

    1. Initial program 55.1

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

    3. Applied flip-+55.1

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

    4. Applied simplify27.0

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

    5. Taylor expanded around inf 17.0

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

    6. Applied simplify6.2

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

    if 5.7722669403334366e-300 < (/ (- 1/2) b_2) < 3.4884196960522e-155

    1. Initial program 60.9

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

    2. Taylor expanded around -inf 1.9

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

    if 3.4884196960522e-155 < (/ (- 1/2) b_2)

    1. Initial program 8.4

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018193 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))