Average Error: 35.9 → 6.7
Time: 1.4m
Precision: 64
Internal Precision: 2944
\[\frac{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b/2 \le -6247282.286903306:\\ \;\;\;\;\frac{c}{b/2} \cdot \frac{-1}{2}\\ \mathbf{if}\;b/2 \le -1.5129849358366514 \cdot 10^{-214}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\left(-b/2\right) + \sqrt{b/2 \cdot b/2 - a \cdot c}}}{a}\\ \mathbf{if}\;b/2 \le 4.022936252353642 \cdot 10^{+122}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b/2}{a}\\ \end{array}\]

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 < -6247282.286903306

    1. Initial program 58.4

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

    2. Taylor expanded around -inf 15.0

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

    3. Applied simplify0.0

      \[\leadsto \color{blue}{\frac{c}{b/2} \cdot \frac{-1}{2}}\]

    if -6247282.286903306 < b/2 < -1.5129849358366514e-214

    1. Initial program 30.2

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

    3. Applied flip--30.3

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

    4. Applied simplify18.4

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

    if -1.5129849358366514e-214 < b/2 < 4.022936252353642e+122

    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 clear-num10.6

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

    if 4.022936252353642e+122 < b/2

    1. Initial program 51.6

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

    2. Taylor expanded around inf 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b/2}{a}}\]
  3. Recombined 4 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed '#(1062803647 245428163 493620569 3595423923 1908391097 2390014376)' 
(FPCore (a b/2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b/2) (sqrt (- (* b/2 b/2) (* a c)))) a))