Average Error: 33.5 → 11.9
Time: 26.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.3725796156555912 \cdot 10^{+127}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 3.207624111695675 \cdot 10^{-187}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{elif}\;b \le 4.664677641347216 \cdot 10^{-111}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 1.922674299151799 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{\frac{b \cdot b - \left(\left(4 \cdot a\right) \cdot c + b \cdot b\right)}{a}}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + b}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -1.3725796156555912 \cdot 10^{+127}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 3.207624111695675 \cdot 10^{-187}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a} - \frac{b}{a}}{2}\\

\mathbf{elif}\;b \le 4.664677641347216 \cdot 10^{-111}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\mathbf{elif}\;b \le 1.922674299151799 \cdot 10^{-16}:\\
\;\;\;\;\frac{\frac{\frac{b \cdot b - \left(\left(4 \cdot a\right) \cdot c + b \cdot b\right)}{a}}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + b}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r1770435 = b;
        double r1770436 = -r1770435;
        double r1770437 = r1770435 * r1770435;
        double r1770438 = 4.0;
        double r1770439 = a;
        double r1770440 = r1770438 * r1770439;
        double r1770441 = c;
        double r1770442 = r1770440 * r1770441;
        double r1770443 = r1770437 - r1770442;
        double r1770444 = sqrt(r1770443);
        double r1770445 = r1770436 + r1770444;
        double r1770446 = 2.0;
        double r1770447 = r1770446 * r1770439;
        double r1770448 = r1770445 / r1770447;
        return r1770448;
}
↓
double f(double a, double b, double c) {
        double r1770449 = b;
        double r1770450 = -1.3725796156555912e+127;
        bool r1770451 = r1770449 <= r1770450;
        double r1770452 = c;
        double r1770453 = r1770452 / r1770449;
        double r1770454 = a;
        double r1770455 = r1770449 / r1770454;
        double r1770456 = r1770453 - r1770455;
        double r1770457 = 2.0;
        double r1770458 = r1770456 * r1770457;
        double r1770459 = r1770458 / r1770457;
        double r1770460 = 3.207624111695675e-187;
        bool r1770461 = r1770449 <= r1770460;
        double r1770462 = r1770449 * r1770449;
        double r1770463 = 4.0;
        double r1770464 = r1770454 * r1770452;
        double r1770465 = r1770463 * r1770464;
        double r1770466 = r1770462 - r1770465;
        double r1770467 = sqrt(r1770466);
        double r1770468 = r1770467 / r1770454;
        double r1770469 = r1770468 - r1770455;
        double r1770470 = r1770469 / r1770457;
        double r1770471 = 4.664677641347216e-111;
        bool r1770472 = r1770449 <= r1770471;
        double r1770473 = -2.0;
        double r1770474 = r1770473 * r1770453;
        double r1770475 = r1770474 / r1770457;
        double r1770476 = 1.922674299151799e-16;
        bool r1770477 = r1770449 <= r1770476;
        double r1770478 = r1770463 * r1770454;
        double r1770479 = r1770478 * r1770452;
        double r1770480 = r1770479 + r1770462;
        double r1770481 = r1770462 - r1770480;
        double r1770482 = r1770481 / r1770454;
        double r1770483 = r1770467 + r1770449;
        double r1770484 = r1770482 / r1770483;
        double r1770485 = r1770484 / r1770457;
        double r1770486 = r1770477 ? r1770485 : r1770475;
        double r1770487 = r1770472 ? r1770475 : r1770486;
        double r1770488 = r1770461 ? r1770470 : r1770487;
        double r1770489 = r1770451 ? r1770459 : r1770488;
        return r1770489;
}

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 < -1.3725796156555912e+127

    1. Initial program 51.4

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

    2. Simplified51.4

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

    3. Taylor expanded around -inf 2.3

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

    4. Simplified2.3

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

    if -1.3725796156555912e+127 < b < 3.207624111695675e-187

    1. Initial program 10.4

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

    2. Simplified10.4

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

    4. Applied div-sub10.4

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

    if 3.207624111695675e-187 < b < 4.664677641347216e-111 or 1.922674299151799e-16 < b

    1. Initial program 50.1

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

    2. Simplified50.1

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

    3. Taylor expanded around inf 12.0

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

    if 4.664677641347216e-111 < b < 1.922674299151799e-16

    1. Initial program 36.3

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

    2. Simplified36.3

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

    4. Applied *-un-lft-identity36.3

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

    5. Applied *-un-lft-identity36.3

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

    6. Applied distribute-lft-out--36.3

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

    7. Applied associate-/l*36.4

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

    9. Applied div-inv36.4

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

    10. Applied associate-/r*36.4

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

    12. Applied flip--36.5

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

    13. Applied associate-/r/36.5

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

    14. Applied associate-/r*36.5

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

    15. Simplified36.4

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

  4. Final simplification11.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3725796156555912 \cdot 10^{+127}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 3.207624111695675 \cdot 10^{-187}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{elif}\;b \le 4.664677641347216 \cdot 10^{-111}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 1.922674299151799 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{\frac{b \cdot b - \left(\left(4 \cdot a\right) \cdot c + b \cdot b\right)}{a}}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + b}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))