Average Error: 33.1 → 9.9
Time: 38.3s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.5978636646744956 \cdot 10^{+151}:\\ \;\;\;\;\frac{(\frac{1}{2} \cdot \left(\frac{a}{\frac{b_2}{c}}\right) + \left(b_2 \cdot -2\right))_*}{a}\\ \mathbf{elif}\;b_2 \le 2.4345792893709955 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.5978636646744956 \cdot 10^{+151}:\\
\;\;\;\;\frac{(\frac{1}{2} \cdot \left(\frac{a}{\frac{b_2}{c}}\right) + \left(b_2 \cdot -2\right))_*}{a}\\

\mathbf{elif}\;b_2 \le 2.4345792893709955 \cdot 10^{-33}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r1093972 = b_2;
        double r1093973 = -r1093972;
        double r1093974 = r1093972 * r1093972;
        double r1093975 = a;
        double r1093976 = c;
        double r1093977 = r1093975 * r1093976;
        double r1093978 = r1093974 - r1093977;
        double r1093979 = sqrt(r1093978);
        double r1093980 = r1093973 + r1093979;
        double r1093981 = r1093980 / r1093975;
        return r1093981;
}
↓
double f(double a, double b_2, double c) {
        double r1093982 = b_2;
        double r1093983 = -1.5978636646744956e+151;
        bool r1093984 = r1093982 <= r1093983;
        double r1093985 = 0.5;
        double r1093986 = a;
        double r1093987 = c;
        double r1093988 = r1093982 / r1093987;
        double r1093989 = r1093986 / r1093988;
        double r1093990 = -2.0;
        double r1093991 = r1093982 * r1093990;
        double r1093992 = fma(r1093985, r1093989, r1093991);
        double r1093993 = r1093992 / r1093986;
        double r1093994 = 2.4345792893709955e-33;
        bool r1093995 = r1093982 <= r1093994;
        double r1093996 = r1093982 * r1093982;
        double r1093997 = r1093986 * r1093987;
        double r1093998 = r1093996 - r1093997;
        double r1093999 = sqrt(r1093998);
        double r1094000 = r1093999 - r1093982;
        double r1094001 = r1094000 / r1093986;
        double r1094002 = -0.5;
        double r1094003 = r1093987 / r1093982;
        double r1094004 = r1094002 * r1094003;
        double r1094005 = r1093995 ? r1094001 : r1094004;
        double r1094006 = r1093984 ? r1093993 : r1094005;
        return r1094006;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -1.5978636646744956e+151

    1. Initial program 59.9

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

    2. Simplified59.9

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

    3. Taylor expanded around -inf 8.9

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

    4. Simplified1.7

      \[\leadsto \frac{\color{blue}{(\frac{1}{2} \cdot \left(\frac{a}{\frac{b_2}{c}}\right) + \left(-2 \cdot b_2\right))_*}}{a}\]

    if -1.5978636646744956e+151 < b_2 < 2.4345792893709955e-33

    1. Initial program 13.6

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

    2. Simplified13.6

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

    3. Taylor expanded around -inf 13.6

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

    4. Simplified13.6

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

    if 2.4345792893709955e-33 < b_2

    1. Initial program 53.7

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

    2. Simplified53.7

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

    3. Taylor expanded around inf 6.9

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.5978636646744956 \cdot 10^{+151}:\\ \;\;\;\;\frac{(\frac{1}{2} \cdot \left(\frac{a}{\frac{b_2}{c}}\right) + \left(b_2 \cdot -2\right))_*}{a}\\ \mathbf{elif}\;b_2 \le 2.4345792893709955 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))