Average Error: 19.0 → 6.9
Time: 20.5s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.665963569877209 \cdot 10^{+64}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \le 2.0167317209326723 \cdot 10^{+50}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\left(b - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - \frac{a \cdot 2}{\frac{b}{c}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}{a \cdot 2}\\ \end{array}\]
\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 -1.665963569877209 \cdot 10^{+64}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{2}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{c}}\\

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

\end{array}\\

\mathbf{elif}\;b \le 2.0167317209326723 \cdot 10^{+50}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\

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

\end{array}\\

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

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

\end{array}
double f(double a, double b, double c) {
        double r559101 = b;
        double r559102 = 0.0;
        bool r559103 = r559101 >= r559102;
        double r559104 = 2.0;
        double r559105 = c;
        double r559106 = r559104 * r559105;
        double r559107 = -r559101;
        double r559108 = r559101 * r559101;
        double r559109 = 4.0;
        double r559110 = a;
        double r559111 = r559109 * r559110;
        double r559112 = r559111 * r559105;
        double r559113 = r559108 - r559112;
        double r559114 = sqrt(r559113);
        double r559115 = r559107 - r559114;
        double r559116 = r559106 / r559115;
        double r559117 = r559107 + r559114;
        double r559118 = r559104 * r559110;
        double r559119 = r559117 / r559118;
        double r559120 = r559103 ? r559116 : r559119;
        return r559120;
}
↓
double f(double a, double b, double c) {
        double r559121 = b;
        double r559122 = -1.665963569877209e+64;
        bool r559123 = r559121 <= r559122;
        double r559124 = 0.0;
        bool r559125 = r559121 >= r559124;
        double r559126 = 2.0;
        double r559127 = -r559121;
        double r559128 = r559121 * r559121;
        double r559129 = a;
        double r559130 = 4.0;
        double r559131 = r559129 * r559130;
        double r559132 = c;
        double r559133 = r559131 * r559132;
        double r559134 = r559128 - r559133;
        double r559135 = sqrt(r559134);
        double r559136 = r559127 - r559135;
        double r559137 = r559136 / r559132;
        double r559138 = r559126 / r559137;
        double r559139 = r559132 / r559121;
        double r559140 = r559121 / r559129;
        double r559141 = r559139 - r559140;
        double r559142 = r559125 ? r559138 : r559141;
        double r559143 = 2.0167317209326723e+50;
        bool r559144 = r559121 <= r559143;
        double r559145 = r559126 * r559132;
        double r559146 = r559145 / r559136;
        double r559147 = r559121 - r559135;
        double r559148 = -r559147;
        double r559149 = r559129 * r559126;
        double r559150 = r559148 / r559149;
        double r559151 = r559125 ? r559146 : r559150;
        double r559152 = r559121 / r559132;
        double r559153 = r559149 / r559152;
        double r559154 = r559121 - r559153;
        double r559155 = r559127 - r559154;
        double r559156 = r559145 / r559155;
        double r559157 = r559129 / r559152;
        double r559158 = r559157 - r559121;
        double r559159 = r559126 * r559158;
        double r559160 = r559159 / r559149;
        double r559161 = r559125 ? r559156 : r559160;
        double r559162 = r559144 ? r559151 : r559161;
        double r559163 = r559123 ? r559142 : r559162;
        return r559163;
}

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 < -1.665963569877209e+64

    1. Initial program 37.7

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

    2. Taylor expanded around -inf 10.6

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

    3. Simplified5.2

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

    4. Taylor expanded around 0 5.2

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

    6. Applied associate-/l*5.2

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

    if -1.665963569877209e+64 < b < 2.0167317209326723e+50

    1. Initial program 9.2

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

    3. Applied add-sqr-sqrt9.3

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

    5. Applied neg-sub09.3

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

    6. Applied associate-+l-9.3

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

    7. Simplified9.2

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

    if 2.0167317209326723e+50 < b

    1. Initial program 25.3

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

    2. Taylor expanded around -inf 25.3

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

    3. Simplified25.3

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

    4. Taylor expanded around inf 7.2

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

    5. Simplified3.9

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

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.665963569877209 \cdot 10^{+64}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \le 2.0167317209326723 \cdot 10^{+50}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\left(b - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - \frac{a \cdot 2}{\frac{b}{c}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}{a \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 
(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))))