Average Error: 19.0 → 7.3
Time: 26.2s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.037202934115589 \cdot 10^{+61}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{a}{b} \cdot c - b\right) \cdot 2}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \le 3.1115579814291686 \cdot 10^{+29}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4\right)}} \cdot \left|\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4\right)}\right|}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{a}{b} \cdot c - b\right) \cdot 2}{a \cdot 2}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 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 -3.037202934115589 \cdot 10^{+61}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}\\

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

\end{array}\\

\mathbf{elif}\;b \le 3.1115579814291686 \cdot 10^{+29}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4\right)}} \cdot \left|\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4\right)}\right|}{a \cdot 2}\\

\end{array}\\

\mathbf{elif}\;b \ge 0:\\
\;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1030059 = b;
        double r1030060 = 0.0;
        bool r1030061 = r1030059 >= r1030060;
        double r1030062 = 2.0;
        double r1030063 = c;
        double r1030064 = r1030062 * r1030063;
        double r1030065 = -r1030059;
        double r1030066 = r1030059 * r1030059;
        double r1030067 = 4.0;
        double r1030068 = a;
        double r1030069 = r1030067 * r1030068;
        double r1030070 = r1030069 * r1030063;
        double r1030071 = r1030066 - r1030070;
        double r1030072 = sqrt(r1030071);
        double r1030073 = r1030065 - r1030072;
        double r1030074 = r1030064 / r1030073;
        double r1030075 = r1030065 + r1030072;
        double r1030076 = r1030062 * r1030068;
        double r1030077 = r1030075 / r1030076;
        double r1030078 = r1030061 ? r1030074 : r1030077;
        return r1030078;
}
↓
double f(double a, double b, double c) {
        double r1030079 = b;
        double r1030080 = -3.037202934115589e+61;
        bool r1030081 = r1030079 <= r1030080;
        double r1030082 = 0.0;
        bool r1030083 = r1030079 >= r1030082;
        double r1030084 = 2.0;
        double r1030085 = c;
        double r1030086 = r1030084 * r1030085;
        double r1030087 = a;
        double r1030088 = r1030079 / r1030085;
        double r1030089 = r1030087 / r1030088;
        double r1030090 = r1030089 - r1030079;
        double r1030091 = r1030084 * r1030090;
        double r1030092 = r1030086 / r1030091;
        double r1030093 = r1030087 / r1030079;
        double r1030094 = r1030093 * r1030085;
        double r1030095 = r1030094 - r1030079;
        double r1030096 = r1030095 * r1030084;
        double r1030097 = r1030087 * r1030084;
        double r1030098 = r1030096 / r1030097;
        double r1030099 = r1030083 ? r1030092 : r1030098;
        double r1030100 = 3.1115579814291686e+29;
        bool r1030101 = r1030079 <= r1030100;
        double r1030102 = -r1030079;
        double r1030103 = r1030079 * r1030079;
        double r1030104 = 4.0;
        double r1030105 = r1030087 * r1030104;
        double r1030106 = r1030085 * r1030105;
        double r1030107 = r1030103 - r1030106;
        double r1030108 = sqrt(r1030107);
        double r1030109 = r1030102 - r1030108;
        double r1030110 = r1030086 / r1030109;
        double r1030111 = cbrt(r1030107);
        double r1030112 = sqrt(r1030111);
        double r1030113 = fabs(r1030111);
        double r1030114 = r1030112 * r1030113;
        double r1030115 = r1030102 + r1030114;
        double r1030116 = r1030115 / r1030097;
        double r1030117 = r1030083 ? r1030110 : r1030116;
        double r1030118 = r1030101 ? r1030117 : r1030099;
        double r1030119 = r1030081 ? r1030099 : r1030118;
        return r1030119;
}

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 2 regimes

  2. if b < -3.037202934115589e+61 or 3.1115579814291686e+29 < b

    1. Initial program 29.2

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt29.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]

    4. Taylor expanded around -inf 18.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2 \cdot a}\\ \end{array}\]

    5. Simplified16.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(c \cdot \frac{a}{b} - b\right)}{2 \cdot a}\\ \end{array}\]

    6. Taylor expanded around inf 7.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(c \cdot \frac{a}{b} - b\right)}{2 \cdot a}\\ \end{array}\]

    7. Simplified4.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(c \cdot \frac{a}{b} - b\right)}{2 \cdot a}\\ \end{array}\]

    if -3.037202934115589e+61 < b < 3.1115579814291686e+29

    1. Initial program 9.3

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt9.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left(\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]

    4. Applied sqrt-prod9.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]

    5. Simplified9.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left|\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right| \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification7.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.037202934115589 \cdot 10^{+61}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{a}{b} \cdot c - b\right) \cdot 2}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \le 3.1115579814291686 \cdot 10^{+29}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4\right)}} \cdot \left|\sqrt[3]{b \cdot b - c \cdot \left(a \cdot 4\right)}\right|}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{a}{b} \cdot c - b\right) \cdot 2}{a \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 2"
  (if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))))