Average Error: 34.2 → 10.4
Time: 10.7s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.12310353364421125 \cdot 10^{95}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 3.446447862996811 \cdot 10^{-75}:\\ \;\;\;\;\frac{-1}{-\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -4.12310353364421125 \cdot 10^{95}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 3.446447862996811 \cdot 10^{-75}:\\
\;\;\;\;\frac{-1}{-\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\end{array}
double f(double a, double b_2, double c) {
        double r17105 = b_2;
        double r17106 = -r17105;
        double r17107 = r17105 * r17105;
        double r17108 = a;
        double r17109 = c;
        double r17110 = r17108 * r17109;
        double r17111 = r17107 - r17110;
        double r17112 = sqrt(r17111);
        double r17113 = r17106 + r17112;
        double r17114 = r17113 / r17108;
        return r17114;
}
↓
double f(double a, double b_2, double c) {
        double r17115 = b_2;
        double r17116 = -4.123103533644211e+95;
        bool r17117 = r17115 <= r17116;
        double r17118 = 0.5;
        double r17119 = c;
        double r17120 = r17119 / r17115;
        double r17121 = r17118 * r17120;
        double r17122 = 2.0;
        double r17123 = a;
        double r17124 = r17115 / r17123;
        double r17125 = r17122 * r17124;
        double r17126 = r17121 - r17125;
        double r17127 = 3.446447862996811e-75;
        bool r17128 = r17115 <= r17127;
        double r17129 = -1.0;
        double r17130 = r17115 * r17115;
        double r17131 = r17123 * r17119;
        double r17132 = r17130 - r17131;
        double r17133 = sqrt(r17132);
        double r17134 = r17133 - r17115;
        double r17135 = r17123 / r17134;
        double r17136 = -r17135;
        double r17137 = r17129 / r17136;
        double r17138 = -0.5;
        double r17139 = r17138 * r17120;
        double r17140 = r17128 ? r17137 : r17139;
        double r17141 = r17117 ? r17126 : r17140;
        return r17141;
}

Error

Bits error versus a

Bits error versus b_2

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_2 < -4.123103533644211e+95

    1. Initial program 47.3

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Simplified47.3

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

    3. Taylor expanded around -inf 3.8

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]

    if -4.123103533644211e+95 < b_2 < 3.446447862996811e-75

    1. Initial program 13.3

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Simplified13.3

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
    3. Using strategy rm

    4. Applied clear-num13.4

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}\]
    5. Using strategy rm

    6. Applied frac-2neg13.4

      \[\leadsto \color{blue}{\frac{-1}{-\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}\]

    7. Simplified13.4

      \[\leadsto \frac{\color{blue}{-1}}{-\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\]

    if 3.446447862996811e-75 < b_2

    1. Initial program 52.5

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Simplified52.5

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

    3. Taylor expanded around inf 9.7

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.12310353364421125 \cdot 10^{95}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 3.446447862996811 \cdot 10^{-75}:\\ \;\;\;\;\frac{-1}{-\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))