Average Error: 34.1 → 8.7
Time: 7.2s
Precision: binary64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.9961865930381599 \cdot 10^{-37}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -9.65316982847353442 \cdot 10^{-258}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}{c}}}{a}\\ \mathbf{elif}\;b_2 \le 1.9796318503870474 \cdot 10^{101}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

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 < -3.9961865930381599e-37

    1. Initial program 55.0

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

    2. Taylor expanded around -inf 7.3

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

    if -3.9961865930381599e-37 < b_2 < -9.65316982847353442e-258

    1. Initial program 23.5

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

    3. Applied flip--23.6

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

    4. Simplified17.5

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

    5. Simplified17.5

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied clear-num17.5

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{0 + a \cdot c}}}}{a}\]

    8. Simplified13.5

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

    if -9.65316982847353442e-258 < b_2 < 1.9796318503870474e101

    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 1.9796318503870474e101 < b_2

    1. Initial program 46.8

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

    2. Taylor expanded around inf 4.1

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.9961865930381599 \cdot 10^{-37}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -9.65316982847353442 \cdot 10^{-258}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}{c}}}{a}\\ \mathbf{elif}\;b_2 \le 1.9796318503870474 \cdot 10^{101}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (neg b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))