Average Error: 34.0 → 9.2
Time: 3.3m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -9.364913351832592 \cdot 10^{+42}:\\ \;\;\;\;\left(-\frac{1}{2}\right) \cdot \frac{c}{b_2}\\ \mathbf{if}\;b_2 \le -4.099879830209986 \cdot 10^{-284}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\\ \mathbf{if}\;b_2 \le 5.317885823358275 \cdot 10^{+67}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array}\]

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 < -9.364913351832592e+42

    1. Initial program 56.7

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

    2. Taylor expanded around -inf 43.2

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

    3. Applied simplify4.8

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

    if -9.364913351832592e+42 < b_2 < -4.099879830209986e-284

    1. Initial program 29.5

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

    3. Applied flip--29.6

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

    4. Applied simplify16.7

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

    5. Applied simplify16.7

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

    if -4.099879830209986e-284 < b_2 < 5.317885823358275e+67

    1. Initial program 10.0

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

    3. Applied div-inv10.1

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

    if 5.317885823358275e+67 < b_2

    1. Initial program 39.6

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

    2. Taylor expanded around inf 5.0

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

  4. Applied simplify9.2

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b_2 \le -9.364913351832592 \cdot 10^{+42}:\\ \;\;\;\;\left(-\frac{1}{2}\right) \cdot \frac{c}{b_2}\\ \mathbf{if}\;b_2 \le -4.099879830209986 \cdot 10^{-284}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\\ \mathbf{if}\;b_2 \le 5.317885823358275 \cdot 10^{+67}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array}}\]

Runtime

Time bar (total: 3.3m)Debug logProfile

herbie shell --seed '#(1072967564 1937075727 894099792 790700740 1036514779 1027793188)' +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))