Average Error: 33.8 → 13.3
Time: 37.0s
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.860057971799534 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{-1}{2} \cdot \frac{a \cdot c}{b_2}}{a}\\ \mathbf{elif}\;b_2 \le -1.5826573427766718 \cdot 10^{-98}:\\ \;\;\;\;\frac{a \cdot c}{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)\right) \cdot a}\\ \mathbf{elif}\;b_2 \le 3.9030138302514986 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - (\left(a \cdot \frac{c}{b_2}\right) \cdot \frac{-1}{2} + b_2)_*}{a}\\ \end{array}\]

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 < -1.860057971799534e+130

    1. Initial program 60.2

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

    2. Taylor expanded around 0 60.2

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

    3. Taylor expanded around -inf 13.9

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

    if -1.860057971799534e+130 < b_2 < -1.5826573427766718e-98

    1. Initial program 43.6

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

    3. Applied flip--43.7

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

    4. Applied associate-/l/45.8

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

    5. Simplified16.9

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

    if -1.5826573427766718e-98 < b_2 < 3.9030138302514986e-14

    1. Initial program 13.8

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

    3. Applied clear-num13.9

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

    if 3.9030138302514986e-14 < b_2

    1. Initial program 30.3

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

    2. Taylor expanded around 0 30.3

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

    3. Taylor expanded around inf 12.9

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

    4. Simplified9.1

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

  4. Final simplification13.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.860057971799534 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{-1}{2} \cdot \frac{a \cdot c}{b_2}}{a}\\ \mathbf{elif}\;b_2 \le -1.5826573427766718 \cdot 10^{-98}:\\ \;\;\;\;\frac{a \cdot c}{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)\right) \cdot a}\\ \mathbf{elif}\;b_2 \le 3.9030138302514986 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - (\left(a \cdot \frac{c}{b_2}\right) \cdot \frac{-1}{2} + b_2)_*}{a}\\ \end{array}\]

Runtime

Time bar (total: 37.0s)Debug logProfile

herbie shell --seed 2018249 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))