Average Error: 33.2 → 8.3
Time: 1.4m
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b/2\right) + \sqrt{b/2 \cdot b/2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b/2 \le -6.381156137519095 \cdot 10^{+117}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b/2} - 2 \cdot \frac{b/2}{a}\\ \mathbf{if}\;b/2 \le 3.1658320264775957 \cdot 10^{-169}:\\ \;\;\;\;\left(\left(-b/2\right) + \sqrt{b/2 \cdot b/2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{if}\;b/2 \le 2.1421475396766052 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b/2} \cdot \frac{-1}{2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b/2

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b/2 < -6.381156137519095e+117

    1. Initial program 48.2

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

    3. Applied div-inv48.2

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

    4. Taylor expanded around -inf 2.7

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

    if -6.381156137519095e+117 < b/2 < 3.1658320264775957e-169

    1. Initial program 9.9

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

    3. Applied div-inv10.0

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

    if 3.1658320264775957e-169 < b/2 < 2.1421475396766052e+88

    1. Initial program 37.1

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

    3. Applied flip-+37.1

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

    4. Applied simplify15.6

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

    if 2.1421475396766052e+88 < b/2

    1. Initial program 58.3

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

    2. Taylor expanded around inf 15.0

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

    3. Applied simplify2.4

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed '#(1063058331 1508344079 3191715834 2470104540 4213459606 1468189912)' 
(FPCore (a b/2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b/2) (sqrt (- (* b/2 b/2) (* a c)))) a))