Average Error: 33.4 → 8.6
Time: 1.2m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{c}{\left(-\frac{1}{2}\right) \cdot \frac{c}{b_2}} \le -3.627628867663558 \cdot 10^{+155}:\\ \;\;\;\;\frac{c \cdot \frac{1}{2}}{b_2} - \frac{2}{\frac{a}{b_2}}\\ \mathbf{if}\;\frac{c}{\left(-\frac{1}{2}\right) \cdot \frac{c}{b_2}} \le 1.448081957215979 \cdot 10^{-308}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{if}\;\frac{c}{\left(-\frac{1}{2}\right) \cdot \frac{c}{b_2}} \le 7.821809361233544 \cdot 10^{+60}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{\frac{1}{2} \cdot c}{\frac{b_2}{a}} - \left(b_2 + b_2\right)}\\ \end{array}\]

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 (/ c (* (- 1/2) (/ c b_2))) < -3.627628867663558e+155

    1. Initial program 53.6

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

    2. Taylor expanded around inf 9.5

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

    3. Applied simplify2.1

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

    if -3.627628867663558e+155 < (/ c (* (- 1/2) (/ c b_2))) < 1.448081957215979e-308

    1. Initial program 10.4

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

    if 1.448081957215979e-308 < (/ c (* (- 1/2) (/ c b_2))) < 7.821809361233544e+60

    1. Initial program 32.0

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

    3. Applied flip--32.1

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

    4. Applied simplify15.3

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

    5. Applied simplify15.3

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

    if 7.821809361233544e+60 < (/ c (* (- 1/2) (/ c b_2)))

    1. Initial program 56.9

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

    3. Applied flip--57.0

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

    4. Applied simplify29.2

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

    5. Applied simplify29.2

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

    6. Taylor expanded around -inf 15.4

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

    7. Applied simplify3.4

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018198 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))