Average Error: 33.6 → 10.4
Time: 18.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.1144981103869975 \cdot 10^{+131}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 4.5810084990875205 \cdot 10^{-68}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -2.1144981103869975 \cdot 10^{+131}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 4.5810084990875205 \cdot 10^{-68}:\\
\;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2365994 = b;
        double r2365995 = -r2365994;
        double r2365996 = r2365994 * r2365994;
        double r2365997 = 4.0;
        double r2365998 = a;
        double r2365999 = r2365997 * r2365998;
        double r2366000 = c;
        double r2366001 = r2365999 * r2366000;
        double r2366002 = r2365996 - r2366001;
        double r2366003 = sqrt(r2366002);
        double r2366004 = r2365995 + r2366003;
        double r2366005 = 2.0;
        double r2366006 = r2366005 * r2365998;
        double r2366007 = r2366004 / r2366006;
        return r2366007;
}
↓
double f(double a, double b, double c) {
        double r2366008 = b;
        double r2366009 = -2.1144981103869975e+131;
        bool r2366010 = r2366008 <= r2366009;
        double r2366011 = c;
        double r2366012 = r2366011 / r2366008;
        double r2366013 = a;
        double r2366014 = r2366008 / r2366013;
        double r2366015 = r2366012 - r2366014;
        double r2366016 = 4.5810084990875205e-68;
        bool r2366017 = r2366008 <= r2366016;
        double r2366018 = 1.0;
        double r2366019 = r2366008 * r2366008;
        double r2366020 = r2366011 * r2366013;
        double r2366021 = 4.0;
        double r2366022 = r2366020 * r2366021;
        double r2366023 = r2366019 - r2366022;
        double r2366024 = sqrt(r2366023);
        double r2366025 = r2366024 - r2366008;
        double r2366026 = 2.0;
        double r2366027 = r2366025 / r2366026;
        double r2366028 = r2366013 / r2366027;
        double r2366029 = r2366018 / r2366028;
        double r2366030 = -r2366012;
        double r2366031 = r2366017 ? r2366029 : r2366030;
        double r2366032 = r2366010 ? r2366015 : r2366031;
        return r2366032;
}

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 < -2.1144981103869975e+131

    1. Initial program 53.8

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

    2. Taylor expanded around -inf 2.6

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

    if -2.1144981103869975e+131 < b < 4.5810084990875205e-68

    1. Initial program 13.3

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

    3. Applied div-inv13.5

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

    4. Simplified13.5

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

    6. Applied associate-*r/13.3

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

    7. Simplified13.3

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

    8. Taylor expanded around 0 13.3

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

    9. Simplified13.3

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

    11. Applied clear-num13.4

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

    if 4.5810084990875205e-68 < b

    1. Initial program 52.0

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

    2. Taylor expanded around inf 9.3

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

    3. Simplified9.3

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.1144981103869975 \cdot 10^{+131}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 4.5810084990875205 \cdot 10^{-68}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))