Average Error: 19.6 → 7.9
Time: 22.3s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.340453357951692868039545417802281002798 \cdot 10^{154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(2 \cdot \frac{a \cdot c}{b} - b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 2.269665110513950885787223322334260534415 \cdot 10^{61}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left|\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right| \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - 2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{b}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(b - \frac{2 \cdot a}{\frac{b}{c}}\right)}{2 \cdot a}\\ \end{array}\]
\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 -1.340453357951692868039545417802281002798 \cdot 10^{154}:\\
\;\;\;\;\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) + \left(2 \cdot \frac{a \cdot c}{b} - b\right)}{2 \cdot a}\\

\end{array}\\

\mathbf{elif}\;b \le 2.269665110513950885787223322334260534415 \cdot 10^{61}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left|\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right| \cdot \sqrt{\sqrt[3]{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}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - 2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{b}}\right)\right)}\\

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

\end{array}
double f(double a, double b, double c) {
        double r45013 = b;
        double r45014 = 0.0;
        bool r45015 = r45013 >= r45014;
        double r45016 = 2.0;
        double r45017 = c;
        double r45018 = r45016 * r45017;
        double r45019 = -r45013;
        double r45020 = r45013 * r45013;
        double r45021 = 4.0;
        double r45022 = a;
        double r45023 = r45021 * r45022;
        double r45024 = r45023 * r45017;
        double r45025 = r45020 - r45024;
        double r45026 = sqrt(r45025);
        double r45027 = r45019 - r45026;
        double r45028 = r45018 / r45027;
        double r45029 = r45019 + r45026;
        double r45030 = r45016 * r45022;
        double r45031 = r45029 / r45030;
        double r45032 = r45015 ? r45028 : r45031;
        return r45032;
}
↓
double f(double a, double b, double c) {
        double r45033 = b;
        double r45034 = -1.3404533579516929e+154;
        bool r45035 = r45033 <= r45034;
        double r45036 = 0.0;
        bool r45037 = r45033 >= r45036;
        double r45038 = 2.0;
        double r45039 = c;
        double r45040 = r45038 * r45039;
        double r45041 = -r45033;
        double r45042 = r45033 * r45033;
        double r45043 = 4.0;
        double r45044 = a;
        double r45045 = r45043 * r45044;
        double r45046 = r45045 * r45039;
        double r45047 = r45042 - r45046;
        double r45048 = sqrt(r45047);
        double r45049 = r45041 - r45048;
        double r45050 = r45040 / r45049;
        double r45051 = r45044 * r45039;
        double r45052 = r45051 / r45033;
        double r45053 = r45038 * r45052;
        double r45054 = r45053 - r45033;
        double r45055 = r45041 + r45054;
        double r45056 = r45038 * r45044;
        double r45057 = r45055 / r45056;
        double r45058 = r45037 ? r45050 : r45057;
        double r45059 = 2.269665110513951e+61;
        bool r45060 = r45033 <= r45059;
        double r45061 = cbrt(r45047);
        double r45062 = fabs(r45061);
        double r45063 = sqrt(r45061);
        double r45064 = r45062 * r45063;
        double r45065 = r45041 - r45064;
        double r45066 = r45040 / r45065;
        double r45067 = r45041 + r45048;
        double r45068 = r45067 / r45056;
        double r45069 = r45037 ? r45066 : r45068;
        double r45070 = cbrt(r45033);
        double r45071 = r45070 * r45070;
        double r45072 = r45044 / r45071;
        double r45073 = r45039 / r45070;
        double r45074 = r45072 * r45073;
        double r45075 = r45038 * r45074;
        double r45076 = r45033 - r45075;
        double r45077 = r45041 - r45076;
        double r45078 = r45040 / r45077;
        double r45079 = r45033 / r45039;
        double r45080 = r45056 / r45079;
        double r45081 = r45033 - r45080;
        double r45082 = r45041 + r45081;
        double r45083 = r45082 / r45056;
        double r45084 = r45037 ? r45078 : r45083;
        double r45085 = r45060 ? r45069 : r45084;
        double r45086 = r45035 ? r45058 : r45085;
        return r45086;
}

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 < -1.3404533579516929e+154

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 11.2

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

    if -1.3404533579516929e+154 < b < 2.269665110513951e+61

    1. Initial program 9.0

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

    3. Applied add-cube-cbrt9.2

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

    4. Applied sqrt-prod9.2

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

    5. Simplified9.2

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

    if 2.269665110513951e+61 < b

    1. Initial program 26.3

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

    2. Taylor expanded around inf 7.2

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

    4. Applied add-cube-cbrt7.2

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

    5. Applied times-frac3.9

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

    6. Taylor expanded around inf 3.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - 2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{b}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(b - 2 \cdot \frac{a \cdot c}{b}\right)}{2 \cdot a}\\ \end{array}\]

    7. Simplified3.9

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

  4. Final simplification7.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.340453357951692868039545417802281002798 \cdot 10^{154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(2 \cdot \frac{a \cdot c}{b} - b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 2.269665110513950885787223322334260534415 \cdot 10^{61}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left|\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right| \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - 2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{b}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(b - \frac{2 \cdot a}{\frac{b}{c}}\right)}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(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))))