Average Error: 33.0 → 10.4
Time: 1.5m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.0373992183972371 \cdot 10^{+38}:\\ \;\;\;\;\left(-\frac{1}{2}\right) \cdot \frac{c}{b_2}\\ \mathbf{if}\;b_2 \le -7.587065739737278 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\\ \mathbf{if}\;b_2 \le -2.2869707272504606 \cdot 10^{-173}:\\ \;\;\;\;\left(-\frac{1}{2}\right) \cdot \frac{c}{b_2}\\ \mathbf{if}\;b_2 \le 3.5962548250302795 \cdot 10^{+78}:\\ \;\;\;\;\left(-\frac{b_2}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

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 < -1.0373992183972371e+38 or -7.587065739737278e-20 < b_2 < -2.2869707272504606e-173

    1. Initial program 49.0

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

    2. Taylor expanded around -inf 47.4

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

    3. Applied simplify13.0

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

    if -1.0373992183972371e+38 < b_2 < -7.587065739737278e-20

    1. Initial program 43.2

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

    3. Applied flip--43.2

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

    4. Applied simplify12.9

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

    5. Applied simplify12.9

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

    if -2.2869707272504606e-173 < b_2 < 3.5962548250302795e+78

    1. Initial program 10.1

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

    3. Applied div-sub10.1

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

    if 3.5962548250302795e+78 < b_2

    1. Initial program 40.5

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

    2. Taylor expanded around inf 3.8

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

  4. Applied simplify10.4

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b_2 \le -1.0373992183972371 \cdot 10^{+38}:\\ \;\;\;\;\left(-\frac{1}{2}\right) \cdot \frac{c}{b_2}\\ \mathbf{if}\;b_2 \le -7.587065739737278 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\\ \mathbf{if}\;b_2 \le -2.2869707272504606 \cdot 10^{-173}:\\ \;\;\;\;\left(-\frac{1}{2}\right) \cdot \frac{c}{b_2}\\ \mathbf{if}\;b_2 \le 3.5962548250302795 \cdot 10^{+78}:\\ \;\;\;\;\left(-\frac{b_2}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}}\]

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed 2018199 +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))