Average Error: 33.8 → 8.1
Time: 30.8s
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.55020281846411 \cdot 10^{+144}:\\ \;\;\;\;\frac{c}{(-2 \cdot b_2 + \left(\left(c \cdot a\right) \cdot \frac{\frac{1}{2}}{b_2}\right))_*}\\ \mathbf{elif}\;b_2 \le -3.0257364350404464 \cdot 10^{-307}:\\ \;\;\;\;\frac{\frac{c}{\sqrt{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}}{\sqrt{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}\\ \mathbf{elif}\;b_2 \le 5.523479422929744 \cdot 10^{+47}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{(b_2 \cdot b_2 + \left(\left(-c\right) \cdot a\right))_*}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array}\]

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 < -1.55020281846411e+144

    1. Initial program 61.9

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

    3. Applied fma-neg61.9

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{(b_2 \cdot b_2 + \left(-a \cdot c\right))_*}}}{a}\]
    4. Using strategy rm

    5. Applied flip--62.0

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

    6. Applied associate-/l/62.0

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

    7. Simplified36.4

      \[\leadsto \frac{\color{blue}{a \cdot c}}{a \cdot \left(\left(-b_2\right) + \sqrt{(b_2 \cdot b_2 + \left(-a \cdot c\right))_*}\right)}\]
    8. Using strategy rm

    9. Applied times-frac35.7

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

    10. Simplified35.7

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

    11. Simplified35.7

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

    12. Taylor expanded around -inf 6.3

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

    13. Simplified6.3

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

    if -1.55020281846411e+144 < b_2 < -3.0257364350404464e-307

    1. Initial program 34.4

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

    3. Applied fma-neg34.4

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{(b_2 \cdot b_2 + \left(-a \cdot c\right))_*}}}{a}\]
    4. Using strategy rm

    5. Applied flip--34.5

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

    6. Applied associate-/l/38.6

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

    7. Simplified20.8

      \[\leadsto \frac{\color{blue}{a \cdot c}}{a \cdot \left(\left(-b_2\right) + \sqrt{(b_2 \cdot b_2 + \left(-a \cdot c\right))_*}\right)}\]
    8. Using strategy rm

    9. Applied times-frac8.9

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

    10. Simplified8.9

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

    11. Simplified8.9

      \[\leadsto 1 \cdot \color{blue}{\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\]
    12. Using strategy rm

    13. Applied add-sqr-sqrt9.2

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

    14. Applied associate-/r*9.3

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

    if -3.0257364350404464e-307 < b_2 < 5.523479422929744e+47

    1. Initial program 9.7

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

    3. Applied fma-neg9.7

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

    if 5.523479422929744e+47 < b_2

    1. Initial program 36.1

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

    3. Applied fma-neg36.1

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{(b_2 \cdot b_2 + \left(-a \cdot c\right))_*}}}{a}\]
    4. Using strategy rm

    5. Applied flip--59.6

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

    6. Applied associate-/l/60.2

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

    7. Simplified60.3

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

    8. Taylor expanded around 0 6.0

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

  4. Final simplification8.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.55020281846411 \cdot 10^{+144}:\\ \;\;\;\;\frac{c}{(-2 \cdot b_2 + \left(\left(c \cdot a\right) \cdot \frac{\frac{1}{2}}{b_2}\right))_*}\\ \mathbf{elif}\;b_2 \le -3.0257364350404464 \cdot 10^{-307}:\\ \;\;\;\;\frac{\frac{c}{\sqrt{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}}{\sqrt{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}\\ \mathbf{elif}\;b_2 \le 5.523479422929744 \cdot 10^{+47}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{(b_2 \cdot b_2 + \left(\left(-c\right) \cdot a\right))_*}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array}\]

Runtime

Time bar (total: 30.8s)Debug logProfile

herbie shell --seed 2018254 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))