Average Error: 33.1 → 8.0
Time: 1.0m
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -6.473972066548491 \cdot 10^{+100}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.554031892664371 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{c}{\sqrt[3]{a}} \cdot \frac{a}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt{b_2 \cdot b_2 - c \cdot a} + \left(-b_2\right)}\\ \mathbf{elif}\;b_2 \le 1.983916337927056 \cdot 10^{+89}:\\ \;\;\;\;\left(-\frac{b_2}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\frac{1}{2} \cdot \left(\frac{a}{\frac{b_2}{c}}\right) + \left(b_2 \cdot -2\right))_*}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -6.473972066548491 \cdot 10^{+100}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -3.554031892664371 \cdot 10^{-133}:\\
\;\;\;\;\frac{\frac{c}{\sqrt[3]{a}} \cdot \frac{a}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt{b_2 \cdot b_2 - c \cdot a} + \left(-b_2\right)}\\

\mathbf{elif}\;b_2 \le 1.983916337927056 \cdot 10^{+89}:\\
\;\;\;\;\left(-\frac{b_2}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{(\frac{1}{2} \cdot \left(\frac{a}{\frac{b_2}{c}}\right) + \left(b_2 \cdot -2\right))_*}{a}\\

\end{array}
double f(double a, double b_2, double c) {
        double r2251968 = b_2;
        double r2251969 = -r2251968;
        double r2251970 = r2251968 * r2251968;
        double r2251971 = a;
        double r2251972 = c;
        double r2251973 = r2251971 * r2251972;
        double r2251974 = r2251970 - r2251973;
        double r2251975 = sqrt(r2251974);
        double r2251976 = r2251969 - r2251975;
        double r2251977 = r2251976 / r2251971;
        return r2251977;
}
↓
double f(double a, double b_2, double c) {
        double r2251978 = b_2;
        double r2251979 = -6.473972066548491e+100;
        bool r2251980 = r2251978 <= r2251979;
        double r2251981 = -0.5;
        double r2251982 = c;
        double r2251983 = r2251982 / r2251978;
        double r2251984 = r2251981 * r2251983;
        double r2251985 = -3.554031892664371e-133;
        bool r2251986 = r2251978 <= r2251985;
        double r2251987 = a;
        double r2251988 = cbrt(r2251987);
        double r2251989 = r2251982 / r2251988;
        double r2251990 = r2251988 * r2251988;
        double r2251991 = r2251987 / r2251990;
        double r2251992 = r2251989 * r2251991;
        double r2251993 = r2251978 * r2251978;
        double r2251994 = r2251982 * r2251987;
        double r2251995 = r2251993 - r2251994;
        double r2251996 = sqrt(r2251995);
        double r2251997 = -r2251978;
        double r2251998 = r2251996 + r2251997;
        double r2251999 = r2251992 / r2251998;
        double r2252000 = 1.983916337927056e+89;
        bool r2252001 = r2251978 <= r2252000;
        double r2252002 = r2251978 / r2251987;
        double r2252003 = -r2252002;
        double r2252004 = r2251996 / r2251987;
        double r2252005 = r2252003 - r2252004;
        double r2252006 = 0.5;
        double r2252007 = r2251978 / r2251982;
        double r2252008 = r2251987 / r2252007;
        double r2252009 = -2.0;
        double r2252010 = r2251978 * r2252009;
        double r2252011 = fma(r2252006, r2252008, r2252010);
        double r2252012 = r2252011 / r2251987;
        double r2252013 = r2252001 ? r2252005 : r2252012;
        double r2252014 = r2251986 ? r2251999 : r2252013;
        double r2252015 = r2251980 ? r2251984 : r2252014;
        return r2252015;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -6.473972066548491e+100

    1. Initial program 58.8

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

    3. Applied div-inv58.8

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

    4. Taylor expanded around -inf 2.3

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

    if -6.473972066548491e+100 < b_2 < -3.554031892664371e-133

    1. Initial program 39.0

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

    3. Applied div-inv39.1

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

    5. Applied flip--39.2

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

    6. Applied associate-*l/39.2

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

    7. Simplified13.1

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

    9. Applied add-cube-cbrt14.0

      \[\leadsto \frac{\frac{a \cdot c}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\]

    10. Applied times-frac10.6

      \[\leadsto \frac{\color{blue}{\frac{a}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{c}{\sqrt[3]{a}}}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\]

    if -3.554031892664371e-133 < b_2 < 1.983916337927056e+89

    1. Initial program 11.5

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

    3. Applied div-sub11.5

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

    if 1.983916337927056e+89 < b_2

    1. Initial program 42.0

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

    2. Taylor expanded around inf 9.6

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

    3. Simplified4.2

      \[\leadsto \frac{\color{blue}{(\frac{1}{2} \cdot \left(\frac{a}{\frac{b_2}{c}}\right) + \left(-2 \cdot b_2\right))_*}}{a}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -6.473972066548491 \cdot 10^{+100}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.554031892664371 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{c}{\sqrt[3]{a}} \cdot \frac{a}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt{b_2 \cdot b_2 - c \cdot a} + \left(-b_2\right)}\\ \mathbf{elif}\;b_2 \le 1.983916337927056 \cdot 10^{+89}:\\ \;\;\;\;\left(-\frac{b_2}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\frac{1}{2} \cdot \left(\frac{a}{\frac{b_2}{c}}\right) + \left(b_2 \cdot -2\right))_*}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))