Average Error: 33.8 → 9.4
Time: 17.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.56950087216670373 \cdot 10^{75}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.4951704352063921 \cdot 10^{-301}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\ \mathbf{elif}\;b \le 2.12540180880083329 \cdot 10^{133}:\\ \;\;\;\;\frac{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -3.56950087216670373 \cdot 10^{75}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.4951704352063921 \cdot 10^{-301}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\

\mathbf{elif}\;b \le 2.12540180880083329 \cdot 10^{133}:\\
\;\;\;\;\frac{\frac{\left(4 \cdot a\right) \cdot c}{\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 r39091 = b;
        double r39092 = -r39091;
        double r39093 = r39091 * r39091;
        double r39094 = 4.0;
        double r39095 = a;
        double r39096 = r39094 * r39095;
        double r39097 = c;
        double r39098 = r39096 * r39097;
        double r39099 = r39093 - r39098;
        double r39100 = sqrt(r39099);
        double r39101 = r39092 + r39100;
        double r39102 = 2.0;
        double r39103 = r39102 * r39095;
        double r39104 = r39101 / r39103;
        return r39104;
}
↓
double f(double a, double b, double c) {
        double r39105 = b;
        double r39106 = -3.5695008721667037e+75;
        bool r39107 = r39105 <= r39106;
        double r39108 = 1.0;
        double r39109 = c;
        double r39110 = r39109 / r39105;
        double r39111 = a;
        double r39112 = r39105 / r39111;
        double r39113 = r39110 - r39112;
        double r39114 = r39108 * r39113;
        double r39115 = 1.495170435206392e-301;
        bool r39116 = r39105 <= r39115;
        double r39117 = 1.0;
        double r39118 = 2.0;
        double r39119 = r39118 * r39111;
        double r39120 = r39105 * r39105;
        double r39121 = 4.0;
        double r39122 = r39121 * r39111;
        double r39123 = r39122 * r39109;
        double r39124 = r39120 - r39123;
        double r39125 = sqrt(r39124);
        double r39126 = r39125 - r39105;
        double r39127 = r39119 / r39126;
        double r39128 = r39117 / r39127;
        double r39129 = 2.1254018088008333e+133;
        bool r39130 = r39105 <= r39129;
        double r39131 = -r39105;
        double r39132 = r39131 - r39125;
        double r39133 = r39123 / r39132;
        double r39134 = r39133 / r39119;
        double r39135 = -1.0;
        double r39136 = r39135 * r39110;
        double r39137 = r39130 ? r39134 : r39136;
        double r39138 = r39116 ? r39128 : r39137;
        double r39139 = r39107 ? r39114 : r39138;
        return r39139;
}

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 < -3.5695008721667037e+75

    1. Initial program 42.1

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

    2. Taylor expanded around -inf 4.0

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

    3. Simplified4.0

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

    if -3.5695008721667037e+75 < b < 1.495170435206392e-301

    1. Initial program 9.4

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

    3. Applied clear-num9.6

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

    4. Simplified9.6

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

    if 1.495170435206392e-301 < b < 2.1254018088008333e+133

    1. Initial program 33.6

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

    3. Applied flip-+33.6

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

    4. Simplified16.6

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

    if 2.1254018088008333e+133 < b

    1. Initial program 61.9

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

    2. Taylor expanded around inf 1.7

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

  4. Final simplification9.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.56950087216670373 \cdot 10^{75}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.4951704352063921 \cdot 10^{-301}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\ \mathbf{elif}\;b \le 2.12540180880083329 \cdot 10^{133}:\\ \;\;\;\;\frac{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))