Average Error: 32.9 → 6.5
Time: 31.5s
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.5574647268420826 \cdot 10^{+106}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le -5.6050339820050224 \cdot 10^{-306}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{elif}\;b_2 \le 8.558214290723834 \cdot 10^{+125}:\\ \;\;\;\;\frac{-c}{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

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 < -4.5574647268420826e+106

    1. Initial program 46.3

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

    2. Initial simplification46.3

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

    3. Taylor expanded around -inf 2.8

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

    if -4.5574647268420826e+106 < b_2 < -5.6050339820050224e-306

    1. Initial program 8.9

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

    2. Initial simplification8.9

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

    4. Applied div-inv9.0

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

    6. Applied associate-*r/8.9

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

    if -5.6050339820050224e-306 < b_2 < 8.558214290723834e+125

    1. Initial program 33.1

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

    2. Initial simplification33.1

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

    4. Applied div-inv33.1

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

    6. Applied flip--33.2

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

    7. Applied associate-*l/33.3

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

    8. Simplified14.6

      \[\leadsto \frac{\color{blue}{\frac{\left(-a \cdot c\right) + 0}{a}}}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}\]

    9. Taylor expanded around -inf 8.3

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

    10. Simplified8.3

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

    if 8.558214290723834e+125 < b_2

    1. Initial program 60.2

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

    2. Initial simplification60.2

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

    4. Applied div-inv60.2

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

    5. Taylor expanded around inf 2.3

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.5574647268420826 \cdot 10^{+106}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le -5.6050339820050224 \cdot 10^{-306}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{elif}\;b_2 \le 8.558214290723834 \cdot 10^{+125}:\\ \;\;\;\;\frac{-c}{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Runtime

Time bar (total: 31.5s)Debug logProfile

herbie shell --seed 2018255 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))