Average Error: 34.4 → 6.8
Time: 6.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.30101840923646093 \cdot 10^{98}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -2.37577225186574925 \cdot 10^{-260}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.66645678090455348 \cdot 10^{68}:\\ \;\;\;\;\frac{1 \cdot \left(\frac{1}{\frac{2}{4}} \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -4.30101840923646093 \cdot 10^{98}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -2.37577225186574925 \cdot 10^{-260}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{elif}\;b \le 6.66645678090455348 \cdot 10^{68}:\\
\;\;\;\;\frac{1 \cdot \left(\frac{1}{\frac{2}{4}} \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r66336 = b;
        double r66337 = -r66336;
        double r66338 = r66336 * r66336;
        double r66339 = 4.0;
        double r66340 = a;
        double r66341 = r66339 * r66340;
        double r66342 = c;
        double r66343 = r66341 * r66342;
        double r66344 = r66338 - r66343;
        double r66345 = sqrt(r66344);
        double r66346 = r66337 + r66345;
        double r66347 = 2.0;
        double r66348 = r66347 * r66340;
        double r66349 = r66346 / r66348;
        return r66349;
}
↓
double f(double a, double b, double c) {
        double r66350 = b;
        double r66351 = -4.301018409236461e+98;
        bool r66352 = r66350 <= r66351;
        double r66353 = 1.0;
        double r66354 = c;
        double r66355 = r66354 / r66350;
        double r66356 = a;
        double r66357 = r66350 / r66356;
        double r66358 = r66355 - r66357;
        double r66359 = r66353 * r66358;
        double r66360 = -2.3757722518657493e-260;
        bool r66361 = r66350 <= r66360;
        double r66362 = -r66350;
        double r66363 = r66350 * r66350;
        double r66364 = 4.0;
        double r66365 = r66364 * r66356;
        double r66366 = r66365 * r66354;
        double r66367 = r66363 - r66366;
        double r66368 = sqrt(r66367);
        double r66369 = r66362 + r66368;
        double r66370 = 1.0;
        double r66371 = 2.0;
        double r66372 = r66371 * r66356;
        double r66373 = r66370 / r66372;
        double r66374 = r66369 * r66373;
        double r66375 = 6.6664567809045535e+68;
        bool r66376 = r66350 <= r66375;
        double r66377 = r66371 / r66364;
        double r66378 = r66370 / r66377;
        double r66379 = r66378 * r66354;
        double r66380 = r66370 * r66379;
        double r66381 = r66362 - r66368;
        double r66382 = r66380 / r66381;
        double r66383 = -1.0;
        double r66384 = r66383 * r66355;
        double r66385 = r66376 ? r66382 : r66384;
        double r66386 = r66361 ? r66374 : r66385;
        double r66387 = r66352 ? r66359 : r66386;
        return r66387;
}

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 < -4.301018409236461e+98

    1. Initial program 47.2

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

    2. Taylor expanded around -inf 3.9

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

    3. Simplified3.9

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

    if -4.301018409236461e+98 < b < -2.3757722518657493e-260

    1. Initial program 8.5

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

    3. Applied div-inv8.7

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

    if -2.3757722518657493e-260 < b < 6.6664567809045535e+68

    1. Initial program 29.1

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

    3. Applied flip-+29.1

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

    4. Simplified16.2

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied clear-num16.4

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

    7. Simplified16.0

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

    9. Applied associate-/r*15.8

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

    10. Simplified9.6

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

    12. Applied *-un-lft-identity9.6

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

    13. Applied add-sqr-sqrt9.6

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

    14. Applied times-frac9.6

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

    15. Applied *-un-lft-identity9.6

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

    16. Applied *-un-lft-identity9.6

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

    17. Applied times-frac9.6

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

    18. Applied add-sqr-sqrt9.6

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

    19. Applied times-frac9.6

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

    20. Applied times-frac9.6

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

    21. Simplified9.6

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

    22. Simplified9.5

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

    if 6.6664567809045535e+68 < b

    1. Initial program 58.7

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

    2. Taylor expanded around inf 3.5

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.30101840923646093 \cdot 10^{98}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -2.37577225186574925 \cdot 10^{-260}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.66645678090455348 \cdot 10^{68}:\\ \;\;\;\;\frac{1 \cdot \left(\frac{1}{\frac{2}{4}} \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))