Average Error: 34.1 → 6.4
Time: 15.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.46337219442650475190360405115215330559 \cdot 10^{111}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -3.104620340175948664814965097441450913118 \cdot 10^{-300}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 5.732888581164670930257747643857376081135 \cdot 10^{134}:\\ \;\;\;\;\frac{\frac{c}{\frac{2}{4}}}{-\left(b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.46337219442650475190360405115215330559 \cdot 10^{111}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -3.104620340175948664814965097441450913118 \cdot 10^{-300}:\\
\;\;\;\;\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 5.732888581164670930257747643857376081135 \cdot 10^{134}:\\
\;\;\;\;\frac{\frac{c}{\frac{2}{4}}}{-\left(b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\\

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

\end{array}
double f(double a, double b, double c) {
        double r47339 = b;
        double r47340 = -r47339;
        double r47341 = r47339 * r47339;
        double r47342 = 4.0;
        double r47343 = a;
        double r47344 = r47342 * r47343;
        double r47345 = c;
        double r47346 = r47344 * r47345;
        double r47347 = r47341 - r47346;
        double r47348 = sqrt(r47347);
        double r47349 = r47340 + r47348;
        double r47350 = 2.0;
        double r47351 = r47350 * r47343;
        double r47352 = r47349 / r47351;
        return r47352;
}
↓
double f(double a, double b, double c) {
        double r47353 = b;
        double r47354 = -2.463372194426505e+111;
        bool r47355 = r47353 <= r47354;
        double r47356 = 1.0;
        double r47357 = c;
        double r47358 = r47357 / r47353;
        double r47359 = a;
        double r47360 = r47353 / r47359;
        double r47361 = r47358 - r47360;
        double r47362 = r47356 * r47361;
        double r47363 = -3.1046203401759487e-300;
        bool r47364 = r47353 <= r47363;
        double r47365 = -r47353;
        double r47366 = r47353 * r47353;
        double r47367 = 4.0;
        double r47368 = r47367 * r47359;
        double r47369 = r47368 * r47357;
        double r47370 = r47366 - r47369;
        double r47371 = sqrt(r47370);
        double r47372 = r47365 + r47371;
        double r47373 = 1.0;
        double r47374 = 2.0;
        double r47375 = r47374 * r47359;
        double r47376 = r47373 / r47375;
        double r47377 = r47372 * r47376;
        double r47378 = 5.732888581164671e+134;
        bool r47379 = r47353 <= r47378;
        double r47380 = r47374 / r47367;
        double r47381 = r47357 / r47380;
        double r47382 = r47353 + r47371;
        double r47383 = -r47382;
        double r47384 = r47381 / r47383;
        double r47385 = -1.0;
        double r47386 = r47385 * r47358;
        double r47387 = r47379 ? r47384 : r47386;
        double r47388 = r47364 ? r47377 : r47387;
        double r47389 = r47355 ? r47362 : r47388;
        return r47389;
}

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 < -2.463372194426505e+111

    1. Initial program 48.7

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

    2. Taylor expanded around -inf 3.0

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

    3. Simplified3.0

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

    if -2.463372194426505e+111 < b < -3.1046203401759487e-300

    1. Initial program 8.4

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

    3. Applied div-inv8.6

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

    if -3.1046203401759487e-300 < b < 5.732888581164671e+134

    1. Initial program 33.9

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

    3. Applied flip-+33.9

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

    4. Simplified16.6

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

    6. Applied clear-num16.8

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

    7. Simplified15.6

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

    9. Applied times-frac15.6

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

    10. Simplified9.1

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

    12. Applied div-inv9.1

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

    13. Simplified8.6

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

    if 5.732888581164671e+134 < b

    1. Initial program 62.2

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

    2. Taylor expanded around inf 1.8

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.46337219442650475190360405115215330559 \cdot 10^{111}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -3.104620340175948664814965097441450913118 \cdot 10^{-300}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 5.732888581164670930257747643857376081135 \cdot 10^{134}:\\ \;\;\;\;\frac{\frac{c}{\frac{2}{4}}}{-\left(b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))