Average Error: 33.8 → 12.9
Time: 27.7s
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.101293750180749 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{-1}{2} \cdot \frac{a \cdot c}{b_2}}{a}\\ \mathbf{elif}\;b_2 \le -1.6881136136793343 \cdot 10^{-90}:\\ \;\;\;\;\frac{a \cdot c}{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)\right) \cdot a}\\ \mathbf{elif}\;b_2 \le 3.02491424793758 \cdot 10^{+47}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{{b_2}^{2} - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -2.101293750180749e+143

    1. Initial program 61.8

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

    2. Initial simplification61.8

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

    3. Taylor expanded around -inf 61.8

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

    4. Taylor expanded around -inf 14.6

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

    if -2.101293750180749e+143 < b_2 < -1.6881136136793343e-90

    1. Initial program 43.4

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

    2. Initial simplification43.4

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

    4. Applied flip--43.5

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

    5. Applied associate-/l/45.9

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

    6. Simplified17.1

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

    if -1.6881136136793343e-90 < b_2 < 3.02491424793758e+47

    1. Initial program 13.6

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

    2. Initial simplification13.6

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

    3. Taylor expanded around -inf 13.6

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

    if 3.02491424793758e+47 < b_2

    1. Initial program 36.1

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

    2. Initial simplification36.1

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

    3. Taylor expanded around -inf 36.1

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

    4. Taylor expanded around inf 6.0

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

  4. Final simplification12.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.101293750180749 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{-1}{2} \cdot \frac{a \cdot c}{b_2}}{a}\\ \mathbf{elif}\;b_2 \le -1.6881136136793343 \cdot 10^{-90}:\\ \;\;\;\;\frac{a \cdot c}{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)\right) \cdot a}\\ \mathbf{elif}\;b_2 \le 3.02491424793758 \cdot 10^{+47}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{{b_2}^{2} - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Runtime

Time bar (total: 27.7s)Debug logProfile

herbie shell --seed 2018254 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))