Average Error: 34.2 → 6.5
Time: 18.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.844003813175822562359270713493973222617 \cdot 10^{119}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -4.193871707188482833811342019428468815697 \cdot 10^{-287}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 2.521192511657275894218075856706322414394 \cdot 10^{99}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.844003813175822562359270713493973222617 \cdot 10^{119}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -4.193871707188482833811342019428468815697 \cdot 10^{-287}:\\
\;\;\;\;\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 2.521192511657275894218075856706322414394 \cdot 10^{99}:\\
\;\;\;\;\frac{2 \cdot c}{\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 r39411 = b;
        double r39412 = -r39411;
        double r39413 = r39411 * r39411;
        double r39414 = 4.0;
        double r39415 = a;
        double r39416 = r39414 * r39415;
        double r39417 = c;
        double r39418 = r39416 * r39417;
        double r39419 = r39413 - r39418;
        double r39420 = sqrt(r39419);
        double r39421 = r39412 + r39420;
        double r39422 = 2.0;
        double r39423 = r39422 * r39415;
        double r39424 = r39421 / r39423;
        return r39424;
}
↓
double f(double a, double b, double c) {
        double r39425 = b;
        double r39426 = -1.8440038131758226e+119;
        bool r39427 = r39425 <= r39426;
        double r39428 = 1.0;
        double r39429 = c;
        double r39430 = r39429 / r39425;
        double r39431 = a;
        double r39432 = r39425 / r39431;
        double r39433 = r39430 - r39432;
        double r39434 = r39428 * r39433;
        double r39435 = -4.193871707188483e-287;
        bool r39436 = r39425 <= r39435;
        double r39437 = -r39425;
        double r39438 = r39425 * r39425;
        double r39439 = 4.0;
        double r39440 = r39439 * r39431;
        double r39441 = r39440 * r39429;
        double r39442 = r39438 - r39441;
        double r39443 = sqrt(r39442);
        double r39444 = r39437 + r39443;
        double r39445 = 1.0;
        double r39446 = 2.0;
        double r39447 = r39446 * r39431;
        double r39448 = r39445 / r39447;
        double r39449 = r39444 * r39448;
        double r39450 = 2.521192511657276e+99;
        bool r39451 = r39425 <= r39450;
        double r39452 = r39446 * r39429;
        double r39453 = r39437 - r39443;
        double r39454 = r39452 / r39453;
        double r39455 = -1.0;
        double r39456 = r39455 * r39430;
        double r39457 = r39451 ? r39454 : r39456;
        double r39458 = r39436 ? r39449 : r39457;
        double r39459 = r39427 ? r39434 : r39458;
        return r39459;
}

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 < -1.8440038131758226e+119

    1. Initial program 51.6

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

    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 -1.8440038131758226e+119 < b < -4.193871707188483e-287

    1. Initial program 8.2

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

    3. Applied div-inv8.4

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

    if -4.193871707188483e-287 < b < 2.521192511657276e+99

    1. Initial program 31.6

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

    3. Applied flip-+31.7

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

    4. Simplified16.7

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

    6. Applied div-inv16.8

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

    8. Applied associate-*l/16.1

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

    9. Simplified16.0

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

    10. Taylor expanded around 0 9.4

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

    if 2.521192511657276e+99 < b

    1. Initial program 59.4

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

    2. Taylor expanded around inf 2.6

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.844003813175822562359270713493973222617 \cdot 10^{119}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -4.193871707188482833811342019428468815697 \cdot 10^{-287}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 2.521192511657275894218075856706322414394 \cdot 10^{99}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))