Average Error: 33.2 → 6.3
Time: 3.7m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.956082950649737 \cdot 10^{+126}:\\ \;\;\;\;\frac{c}{b} - \frac{b + b}{2 \cdot a}\\ \mathbf{if}\;b \le 2.2234292265379327 \cdot 10^{-306}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}{2 \cdot a}\\ \mathbf{if}\;b \le 4.718509050094739 \cdot 10^{+75}:\\ \;\;\;\;\frac{4}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}} \cdot \frac{c}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

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 < -9.956082950649737e+126

    1. Initial program 50.8

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

    2. Taylor expanded around -inf 10.5

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

    3. Applied simplify2.9

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

    if -9.956082950649737e+126 < b < 2.2234292265379327e-306

    1. Initial program 9.0

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

    if 2.2234292265379327e-306 < b < 4.718509050094739e+75

    1. Initial program 29.2

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

    3. Applied flip-+29.2

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

    4. Applied simplify15.6

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

    6. Applied *-un-lft-identity15.6

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

    7. Applied times-frac14.2

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

    8. Applied times-frac9.4

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

    9. Applied simplify9.4

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

    10. Applied simplify8.4

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

    if 4.718509050094739e+75 < b

    1. Initial program 57.9

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

    2. Taylor expanded around inf 41.6

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

    3. Applied simplify2.6

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

  4. Applied simplify6.3

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \le -9.956082950649737 \cdot 10^{+126}:\\ \;\;\;\;\frac{c}{b} - \frac{b + b}{2 \cdot a}\\ \mathbf{if}\;b \le 2.2234292265379327 \cdot 10^{-306}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}{2 \cdot a}\\ \mathbf{if}\;b \le 4.718509050094739 \cdot 10^{+75}:\\ \;\;\;\;\frac{4}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}} \cdot \frac{c}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}}\]

Runtime

Time bar (total: 3.7m)Debug logProfile

herbie shell --seed 2018199 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))