Average Error: 32.9 → 10.1
Time: 16.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.7512236628315378 \cdot 10^{+131}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.489031291672483 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -1.7512236628315378 \cdot 10^{+131}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 1.489031291672483 \cdot 10^{-98}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a} - b}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r886995 = b;
        double r886996 = -r886995;
        double r886997 = r886995 * r886995;
        double r886998 = 4.0;
        double r886999 = a;
        double r887000 = r886998 * r886999;
        double r887001 = c;
        double r887002 = r887000 * r887001;
        double r887003 = r886997 - r887002;
        double r887004 = sqrt(r887003);
        double r887005 = r886996 + r887004;
        double r887006 = 2.0;
        double r887007 = r887006 * r886999;
        double r887008 = r887005 / r887007;
        return r887008;
}
↓
double f(double a, double b, double c) {
        double r887009 = b;
        double r887010 = -1.7512236628315378e+131;
        bool r887011 = r887009 <= r887010;
        double r887012 = c;
        double r887013 = r887012 / r887009;
        double r887014 = a;
        double r887015 = r887009 / r887014;
        double r887016 = r887013 - r887015;
        double r887017 = 2.0;
        double r887018 = r887016 * r887017;
        double r887019 = r887018 / r887017;
        double r887020 = 1.489031291672483e-98;
        bool r887021 = r887009 <= r887020;
        double r887022 = r887009 * r887009;
        double r887023 = 4.0;
        double r887024 = r887023 * r887012;
        double r887025 = r887024 * r887014;
        double r887026 = r887022 - r887025;
        double r887027 = sqrt(r887026);
        double r887028 = r887027 - r887009;
        double r887029 = r887028 / r887014;
        double r887030 = r887029 / r887017;
        double r887031 = -2.0;
        double r887032 = r887031 * r887013;
        double r887033 = r887032 / r887017;
        double r887034 = r887021 ? r887030 : r887033;
        double r887035 = r887011 ? r887019 : r887034;
        return r887035;
}

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 < -1.7512236628315378e+131

    1. Initial program 51.5

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

    2. Simplified51.5

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

    3. Taylor expanded around -inf 3.0

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

    4. Simplified3.0

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

    if -1.7512236628315378e+131 < b < 1.489031291672483e-98

    1. Initial program 11.5

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

    2. Simplified11.6

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

    4. Applied *-un-lft-identity11.6

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

    5. Applied associate-/r*11.6

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

    6. Simplified11.6

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

    if 1.489031291672483e-98 < b

    1. Initial program 51.5

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

    2. Simplified51.5

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

    3. Taylor expanded around inf 10.7

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.7512236628315378 \cdot 10^{+131}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.489031291672483 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))