Average Error: 33.4 → 8.2
Time: 8.1s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.060976138917674342180206539993786896862 \cdot 10^{65}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -2.405136408332622646590056830947601333093 \cdot 10^{-287}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{elif}\;b_2 \le 1.649808383145249284359134623845486246162 \cdot 10^{71}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}{c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.060976138917674342180206539993786896862 \cdot 10^{65}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -2.405136408332622646590056830947601333093 \cdot 10^{-287}:\\
\;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\

\mathbf{elif}\;b_2 \le 1.649808383145249284359134623845486246162 \cdot 10^{71}:\\
\;\;\;\;\frac{\frac{1}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}{c}}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r28983 = b_2;
        double r28984 = -r28983;
        double r28985 = r28983 * r28983;
        double r28986 = a;
        double r28987 = c;
        double r28988 = r28986 * r28987;
        double r28989 = r28985 - r28988;
        double r28990 = sqrt(r28989);
        double r28991 = r28984 + r28990;
        double r28992 = r28991 / r28986;
        return r28992;
}
↓
double f(double a, double b_2, double c) {
        double r28993 = b_2;
        double r28994 = -3.0609761389176743e+65;
        bool r28995 = r28993 <= r28994;
        double r28996 = 0.5;
        double r28997 = c;
        double r28998 = r28997 / r28993;
        double r28999 = r28996 * r28998;
        double r29000 = 2.0;
        double r29001 = a;
        double r29002 = r28993 / r29001;
        double r29003 = r29000 * r29002;
        double r29004 = r28999 - r29003;
        double r29005 = -2.4051364083326226e-287;
        bool r29006 = r28993 <= r29005;
        double r29007 = 1.0;
        double r29008 = r28993 * r28993;
        double r29009 = r29001 * r28997;
        double r29010 = r29008 - r29009;
        double r29011 = sqrt(r29010);
        double r29012 = r29011 - r28993;
        double r29013 = r29001 / r29012;
        double r29014 = r29007 / r29013;
        double r29015 = 1.6498083831452493e+71;
        bool r29016 = r28993 <= r29015;
        double r29017 = -r28993;
        double r29018 = r29017 - r29011;
        double r29019 = r29018 / r29001;
        double r29020 = r29019 / r28997;
        double r29021 = r29007 / r29020;
        double r29022 = r29021 / r29001;
        double r29023 = -0.5;
        double r29024 = r29023 * r28998;
        double r29025 = r29016 ? r29022 : r29024;
        double r29026 = r29006 ? r29014 : r29025;
        double r29027 = r28995 ? r29004 : r29026;
        return r29027;
}

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 4 regimes

  2. if b_2 < -3.0609761389176743e+65

    1. Initial program 40.1

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

    2. Taylor expanded around -inf 4.4

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

    if -3.0609761389176743e+65 < b_2 < -2.4051364083326226e-287

    1. Initial program 8.6

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied clear-num8.7

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

    4. Simplified8.7

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

    if -2.4051364083326226e-287 < b_2 < 1.6498083831452493e+71

    1. Initial program 29.6

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied flip-+29.6

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

    4. Simplified15.9

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]
    5. Using strategy rm

    6. Applied clear-num16.1

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{0 + a \cdot c}}}}{a}\]

    7. Simplified14.4

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

    if 1.6498083831452493e+71 < b_2

    1. Initial program 58.0

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

    2. Taylor expanded around inf 3.3

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.060976138917674342180206539993786896862 \cdot 10^{65}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -2.405136408332622646590056830947601333093 \cdot 10^{-287}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{elif}\;b_2 \le 1.649808383145249284359134623845486246162 \cdot 10^{71}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}{c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))