Average Error: 34.6 → 9.8
Time: 4.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.9358923729233266 \cdot 10^{149}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 9.39036747108992214 \cdot 10^{-69}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.9358923729233266 \cdot 10^{149}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 9.39036747108992214 \cdot 10^{-69}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r48500 = b;
        double r48501 = -r48500;
        double r48502 = r48500 * r48500;
        double r48503 = 4.0;
        double r48504 = a;
        double r48505 = r48503 * r48504;
        double r48506 = c;
        double r48507 = r48505 * r48506;
        double r48508 = r48502 - r48507;
        double r48509 = sqrt(r48508);
        double r48510 = r48501 + r48509;
        double r48511 = 2.0;
        double r48512 = r48511 * r48504;
        double r48513 = r48510 / r48512;
        return r48513;
}
↓
double f(double a, double b, double c) {
        double r48514 = b;
        double r48515 = -2.9358923729233266e+149;
        bool r48516 = r48514 <= r48515;
        double r48517 = 1.0;
        double r48518 = c;
        double r48519 = r48518 / r48514;
        double r48520 = a;
        double r48521 = r48514 / r48520;
        double r48522 = r48519 - r48521;
        double r48523 = r48517 * r48522;
        double r48524 = 9.390367471089922e-69;
        bool r48525 = r48514 <= r48524;
        double r48526 = -r48514;
        double r48527 = r48514 * r48514;
        double r48528 = 4.0;
        double r48529 = r48528 * r48520;
        double r48530 = r48529 * r48518;
        double r48531 = r48527 - r48530;
        double r48532 = sqrt(r48531);
        double r48533 = r48526 + r48532;
        double r48534 = 2.0;
        double r48535 = r48534 * r48520;
        double r48536 = r48533 / r48535;
        double r48537 = -1.0;
        double r48538 = r48537 * r48519;
        double r48539 = r48525 ? r48536 : r48538;
        double r48540 = r48516 ? r48523 : r48539;
        return r48540;
}

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 3 regimes

  2. if b < -2.9358923729233266e+149

    1. Initial program 62.1

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

    2. Taylor expanded around -inf 1.7

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

    3. Simplified1.7

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

    if -2.9358923729233266e+149 < b < 9.390367471089922e-69

    1. Initial program 12.5

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

    3. Applied div-inv12.7

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

    5. Applied un-div-inv12.5

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

    if 9.390367471089922e-69 < b

    1. Initial program 53.5

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

    2. Taylor expanded around inf 8.7

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.9358923729233266 \cdot 10^{149}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 9.39036747108992214 \cdot 10^{-69}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))