Average Error: 33.8 → 8.8
Time: 2.0m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.9979183584340477 \cdot 10^{+120}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{if}\;b_2 \le 5.254270674726003 \cdot 10^{-210}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{if}\;b_2 \le 9.988998546709026 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{(\left(\frac{a}{b_2}\right) \cdot \left(\frac{1}{2} \cdot c\right) + \left(\left(-b_2\right) + \left(-b_2\right)\right))_*}\\ \end{array}\]

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 < -1.9979183584340477e+120

    1. Initial program 50.2

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

    2. Taylor expanded around -inf 2.7

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

    if -1.9979183584340477e+120 < b_2 < 5.254270674726003e-210

    1. Initial program 10.5

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

    3. Applied div-inv10.6

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

    if 5.254270674726003e-210 < b_2 < 9.988998546709026e+36

    1. Initial program 32.4

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

    3. Applied flip-+32.5

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

    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 9.988998546709026e+36 < b_2

    1. Initial program 56.3

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

    3. Applied flip-+56.4

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

    4. Applied simplify27.4

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

    5. Taylor expanded around inf 14.9

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

    6. Applied simplify3.8

      \[\leadsto \color{blue}{\frac{c}{(\left(\frac{a}{b_2}\right) \cdot \left(\frac{1}{2} \cdot c\right) + \left(\left(-b_2\right) + \left(-b_2\right)\right))_*}}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed 2020178 +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))