Average Error: 33.7 → 7.3
Time: 1.2m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.4179539573507906 \cdot 10^{+153}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \le -2.689070205918387 \cdot 10^{-289}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.269164629607323 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{c \cdot 4}{2}}{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot 4}{2}}{\left(-b\right) - \left(b - 2 \cdot \frac{c \cdot a}{b}\right)}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

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 < -2.4179539573507906e+153

    1. Initial program 60.6

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Taylor expanded around -inf 2.2

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}}\]

    3. Simplified2.2

      \[\leadsto \color{blue}{\frac{-b}{a}}\]

    if -2.4179539573507906e+153 < b < -2.689070205918387e-289

    1. Initial program 7.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    if -2.689070205918387e-289 < b < 6.269164629607323e+152

    1. Initial program 33.4

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied flip-+33.5

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

    4. Applied associate-/l/37.4

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\]

    5. Simplified20.0

      \[\leadsto \frac{\color{blue}{4 \cdot \left(c \cdot a\right)}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\]
    6. Using strategy rm

    7. Applied times-frac16.4

      \[\leadsto \color{blue}{\frac{4}{2 \cdot a} \cdot \frac{c \cdot a}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\]
    8. Using strategy rm

    9. Applied pow116.4

      \[\leadsto \frac{4}{2 \cdot a} \cdot \color{blue}{{\left(\frac{c \cdot a}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}^{1}}\]

    10. Applied pow116.4

      \[\leadsto \color{blue}{{\left(\frac{4}{2 \cdot a}\right)}^{1}} \cdot {\left(\frac{c \cdot a}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}^{1}\]

    11. Applied pow-prod-down16.4

      \[\leadsto \color{blue}{{\left(\frac{4}{2 \cdot a} \cdot \frac{c \cdot a}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}^{1}}\]

    12. Simplified8.5

      \[\leadsto {\color{blue}{\left(\frac{\frac{c \cdot 4}{\frac{2}{1}}}{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}\right)}}^{1}\]

    if 6.269164629607323e+152 < b

    1. Initial program 62.7

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied flip-+62.7

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

    4. Applied associate-/l/62.7

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\]

    5. Simplified37.2

      \[\leadsto \frac{\color{blue}{4 \cdot \left(c \cdot a\right)}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\]
    6. Using strategy rm

    7. Applied times-frac37.0

      \[\leadsto \color{blue}{\frac{4}{2 \cdot a} \cdot \frac{c \cdot a}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\]
    8. Using strategy rm

    9. Applied pow137.0

      \[\leadsto \frac{4}{2 \cdot a} \cdot \color{blue}{{\left(\frac{c \cdot a}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}^{1}}\]

    10. Applied pow137.0

      \[\leadsto \color{blue}{{\left(\frac{4}{2 \cdot a}\right)}^{1}} \cdot {\left(\frac{c \cdot a}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}^{1}\]

    11. Applied pow-prod-down37.0

      \[\leadsto \color{blue}{{\left(\frac{4}{2 \cdot a} \cdot \frac{c \cdot a}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}^{1}}\]

    12. Simplified36.9

      \[\leadsto {\color{blue}{\left(\frac{\frac{c \cdot 4}{\frac{2}{1}}}{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}\right)}}^{1}\]

    13. Taylor expanded around inf 6.8

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

  4. Final simplification7.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.4179539573507906 \cdot 10^{+153}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \le -2.689070205918387 \cdot 10^{-289}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.269164629607323 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{c \cdot 4}{2}}{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot 4}{2}}{\left(-b\right) - \left(b - 2 \cdot \frac{c \cdot a}{b}\right)}\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018221 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))