Average Error: 36.3 → 6.1
Time: 32.7s
Precision: 64
Internal precision: 2944
\[\frac{\left(-b/2\right) + \sqrt{b/2 \cdot b/2 - a \cdot c}}{a}\]
⬇
\[\begin{array}{l} \mathbf{if}\;b/2 \le -1.83182834227653 \cdot 10^{+137}:\\ \;\;\;\;\frac{\frac{\frac{1}{2} \cdot c}{\frac{b/2}{a}} - \left(b/2 + b/2\right)}{a}\\ \mathbf{if}\;b/2 \le 1.9148583814984746 \cdot 10^{-58}:\\ \;\;\;\;\left(\left(-b/2\right) + \sqrt{b/2 \cdot b/2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b/2\right) + b/2}{a} - \frac{c}{b/2} \cdot \frac{1}{2}\\ \end{array}\]

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 < -1.83182834227653e+137

    1. Initial program 55.2

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

    3. Applied div-inv 55.2

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

    4. Taylor expanded around -inf 14.0

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

    5. Applied simplify 1.7

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

    if -1.83182834227653e+137 < b/2 < 1.9148583814984746e-58

    1. Initial program 12.4

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

    3. Applied div-inv 12.6

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

    if 1.9148583814984746e-58 < b/2

    1. Initial program 58.1

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

    2. Taylor expanded around inf 39.3

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

    3. Applied simplify 0.0

      \[\leadsto \color{blue}{\frac{\left(-b/2\right) + b/2}{a} - \frac{c}{b/2} \cdot \frac{1}{2}}\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 32.7s) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(2329929097 3210370195 3111198779 2406363002 3511342718 2136436390)'
(FPCore (a b/2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b/2) (sqrt (- (* b/2 b/2) (* a c)))) a))