Average Error: 19.3 → 8.7
Time: 12.2s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.738180479687524379069024322817742160506 \cdot 10^{54}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 8.13128291297911797546814361239497721035 \cdot 10^{93}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - 2 \cdot \frac{a \cdot c}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le -5.738180479687524379069024322817742160506 \cdot 10^{54}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

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

\end{array}\\

\mathbf{elif}\;b \le 8.13128291297911797546814361239497721035 \cdot 10^{93}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}{2 \cdot a}\\

\end{array}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - 2 \cdot \frac{a \cdot c}{b}\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\

\end{array}
double f(double a, double b, double c) {
        double r36002 = b;
        double r36003 = 0.0;
        bool r36004 = r36002 >= r36003;
        double r36005 = 2.0;
        double r36006 = c;
        double r36007 = r36005 * r36006;
        double r36008 = -r36002;
        double r36009 = r36002 * r36002;
        double r36010 = 4.0;
        double r36011 = a;
        double r36012 = r36010 * r36011;
        double r36013 = r36012 * r36006;
        double r36014 = r36009 - r36013;
        double r36015 = sqrt(r36014);
        double r36016 = r36008 - r36015;
        double r36017 = r36007 / r36016;
        double r36018 = r36008 + r36015;
        double r36019 = r36005 * r36011;
        double r36020 = r36018 / r36019;
        double r36021 = r36004 ? r36017 : r36020;
        return r36021;
}
↓
double f(double a, double b, double c) {
        double r36022 = b;
        double r36023 = -5.7381804796875244e+54;
        bool r36024 = r36022 <= r36023;
        double r36025 = 0.0;
        bool r36026 = r36022 >= r36025;
        double r36027 = 2.0;
        double r36028 = c;
        double r36029 = r36027 * r36028;
        double r36030 = -r36022;
        double r36031 = r36022 * r36022;
        double r36032 = 4.0;
        double r36033 = a;
        double r36034 = r36032 * r36033;
        double r36035 = r36034 * r36028;
        double r36036 = r36031 - r36035;
        double r36037 = sqrt(r36036);
        double r36038 = r36030 - r36037;
        double r36039 = r36029 / r36038;
        double r36040 = r36033 * r36028;
        double r36041 = r36040 / r36022;
        double r36042 = r36027 * r36041;
        double r36043 = 2.0;
        double r36044 = r36043 * r36022;
        double r36045 = r36042 - r36044;
        double r36046 = r36027 * r36033;
        double r36047 = r36045 / r36046;
        double r36048 = r36026 ? r36039 : r36047;
        double r36049 = 8.131282912979118e+93;
        bool r36050 = r36022 <= r36049;
        double r36051 = r36037 - r36022;
        double r36052 = sqrt(r36051);
        double r36053 = r36052 * r36052;
        double r36054 = r36053 / r36046;
        double r36055 = r36026 ? r36039 : r36054;
        double r36056 = r36022 - r36042;
        double r36057 = r36030 - r36056;
        double r36058 = r36029 / r36057;
        double r36059 = r36051 / r36046;
        double r36060 = r36026 ? r36058 : r36059;
        double r36061 = r36050 ? r36055 : r36060;
        double r36062 = r36024 ? r36048 : r36061;
        return r36062;
}

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 < -5.7381804796875244e+54

    1. Initial program 38.2

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    2. Simplified38.2

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \end{array}}\]

    3. Taylor expanded around -inf 10.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2 \cdot a}\\ \end{array}\]

    if -5.7381804796875244e+54 < b < 8.131282912979118e+93

    1. Initial program 8.8

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    2. Simplified8.8

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \end{array}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt8.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}{2 \cdot a}\\ \end{array}\]

    if 8.131282912979118e+93 < b

    1. Initial program 29.5

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    2. Simplified29.5

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \end{array}}\]

    3. Taylor expanded around inf 6.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.738180479687524379069024322817742160506 \cdot 10^{54}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 8.13128291297911797546814361239497721035 \cdot 10^{93}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - 2 \cdot \frac{a \cdot c}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))))