Average Error: 32.9 → 6.4
Time: 58.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.880394710329243 \cdot 10^{+120}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 5.818192251940127 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 6.6006279600139335 \cdot 10^{+131}:\\ \;\;\;\;\left(c \cdot -2\right) \cdot \frac{1}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -2.880394710329243 \cdot 10^{+120}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 5.818192251940127 \cdot 10^{-227}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a \cdot 2}\\

\mathbf{elif}\;b \le 6.6006279600139335 \cdot 10^{+131}:\\
\;\;\;\;\left(c \cdot -2\right) \cdot \frac{1}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}\\

\mathbf{else}:\\
\;\;\;\;-\frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r9325292 = b;
        double r9325293 = -r9325292;
        double r9325294 = r9325292 * r9325292;
        double r9325295 = 4.0;
        double r9325296 = a;
        double r9325297 = r9325295 * r9325296;
        double r9325298 = c;
        double r9325299 = r9325297 * r9325298;
        double r9325300 = r9325294 - r9325299;
        double r9325301 = sqrt(r9325300);
        double r9325302 = r9325293 + r9325301;
        double r9325303 = 2.0;
        double r9325304 = r9325303 * r9325296;
        double r9325305 = r9325302 / r9325304;
        return r9325305;
}
↓
double f(double a, double b, double c) {
        double r9325306 = b;
        double r9325307 = -2.880394710329243e+120;
        bool r9325308 = r9325306 <= r9325307;
        double r9325309 = c;
        double r9325310 = r9325309 / r9325306;
        double r9325311 = a;
        double r9325312 = r9325306 / r9325311;
        double r9325313 = r9325310 - r9325312;
        double r9325314 = 5.818192251940127e-227;
        bool r9325315 = r9325306 <= r9325314;
        double r9325316 = r9325306 * r9325306;
        double r9325317 = r9325309 * r9325311;
        double r9325318 = 4.0;
        double r9325319 = r9325317 * r9325318;
        double r9325320 = r9325316 - r9325319;
        double r9325321 = sqrt(r9325320);
        double r9325322 = r9325321 - r9325306;
        double r9325323 = 2.0;
        double r9325324 = r9325311 * r9325323;
        double r9325325 = r9325322 / r9325324;
        double r9325326 = 6.6006279600139335e+131;
        bool r9325327 = r9325306 <= r9325326;
        double r9325328 = -2.0;
        double r9325329 = r9325309 * r9325328;
        double r9325330 = 1.0;
        double r9325331 = r9325321 + r9325306;
        double r9325332 = r9325330 / r9325331;
        double r9325333 = r9325329 * r9325332;
        double r9325334 = -r9325310;
        double r9325335 = r9325327 ? r9325333 : r9325334;
        double r9325336 = r9325315 ? r9325325 : r9325335;
        double r9325337 = r9325308 ? r9325313 : r9325336;
        return r9325337;
}

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 < -2.880394710329243e+120

    1. Initial program 49.1

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

    2. Simplified49.1

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

    3. Taylor expanded around -inf 2.7

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

    if -2.880394710329243e+120 < b < 5.818192251940127e-227

    1. Initial program 9.3

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

    2. Simplified9.3

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

    if 5.818192251940127e-227 < b < 6.6006279600139335e+131

    1. Initial program 35.9

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

    2. Simplified35.9

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

    4. Applied *-un-lft-identity35.9

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

    5. Applied *-un-lft-identity35.9

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

    6. Applied distribute-lft-out--35.9

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

    7. Applied associate-/l*35.9

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

    9. Applied flip--36.0

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

    10. Applied associate-/r/36.1

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

    11. Applied *-un-lft-identity36.1

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

    12. Applied times-frac36.1

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

    13. Simplified13.6

      \[\leadsto \color{blue}{\frac{0 - \left(c \cdot a\right) \cdot 4}{\frac{a}{\frac{1}{2}}}} \cdot \frac{1}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}\]

    14. Taylor expanded around inf 7.3

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

    if 6.6006279600139335e+131 < b

    1. Initial program 60.5

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

    2. Simplified60.5

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

    3. Taylor expanded around inf 2.2

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

    4. Simplified2.2

      \[\leadsto \color{blue}{-\frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.880394710329243 \cdot 10^{+120}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 5.818192251940127 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 6.6006279600139335 \cdot 10^{+131}:\\ \;\;\;\;\left(c \cdot -2\right) \cdot \frac{1}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))