Average Error: 33.1 → 9.3
Time: 1.0m
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.8719024022082672 \cdot 10^{+146}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 5.524249207244028 \cdot 10^{-127}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 5.890069572265506 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{a \cdot c}{-a \cdot c} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.8719024022082672 \cdot 10^{+146}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 5.524249207244028 \cdot 10^{-127}:\\
\;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\

\mathbf{elif}\;b_2 \le 5.890069572265506 \cdot 10^{-16}:\\
\;\;\;\;\frac{\frac{a \cdot c}{-a \cdot c} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2\right)}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r6751902 = b_2;
        double r6751903 = -r6751902;
        double r6751904 = r6751902 * r6751902;
        double r6751905 = a;
        double r6751906 = c;
        double r6751907 = r6751905 * r6751906;
        double r6751908 = r6751904 - r6751907;
        double r6751909 = sqrt(r6751908);
        double r6751910 = r6751903 - r6751909;
        double r6751911 = r6751910 / r6751905;
        return r6751911;
}
↓
double f(double a, double b_2, double c) {
        double r6751912 = b_2;
        double r6751913 = -1.8719024022082672e+146;
        bool r6751914 = r6751912 <= r6751913;
        double r6751915 = -0.5;
        double r6751916 = c;
        double r6751917 = r6751916 / r6751912;
        double r6751918 = r6751915 * r6751917;
        double r6751919 = 5.524249207244028e-127;
        bool r6751920 = r6751912 <= r6751919;
        double r6751921 = r6751912 * r6751912;
        double r6751922 = a;
        double r6751923 = r6751922 * r6751916;
        double r6751924 = r6751921 - r6751923;
        double r6751925 = sqrt(r6751924);
        double r6751926 = r6751925 - r6751912;
        double r6751927 = r6751916 / r6751926;
        double r6751928 = 5.890069572265506e-16;
        bool r6751929 = r6751912 <= r6751928;
        double r6751930 = -r6751923;
        double r6751931 = r6751923 / r6751930;
        double r6751932 = r6751925 + r6751912;
        double r6751933 = r6751931 * r6751932;
        double r6751934 = r6751933 / r6751922;
        double r6751935 = 0.5;
        double r6751936 = r6751917 * r6751935;
        double r6751937 = 2.0;
        double r6751938 = r6751912 / r6751922;
        double r6751939 = r6751937 * r6751938;
        double r6751940 = r6751936 - r6751939;
        double r6751941 = r6751929 ? r6751934 : r6751940;
        double r6751942 = r6751920 ? r6751927 : r6751941;
        double r6751943 = r6751914 ? r6751918 : r6751942;
        return r6751943;
}

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 < -1.8719024022082672e+146

    1. Initial program 62.0

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

    2. Taylor expanded around -inf 1.7

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

    if -1.8719024022082672e+146 < b_2 < 5.524249207244028e-127

    1. Initial program 29.1

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

    3. Applied flip--30.1

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

    4. Simplified16.2

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

    5. Simplified16.2

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

    7. Applied *-un-lft-identity16.2

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

    8. Applied *-un-lft-identity16.2

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

    9. Applied times-frac16.2

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

    10. Simplified16.2

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

    11. Simplified10.4

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

    if 5.524249207244028e-127 < b_2 < 5.890069572265506e-16

    1. Initial program 6.4

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

    3. Applied flip--36.2

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

    4. Simplified36.2

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

    5. Simplified36.2

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

    7. Applied flip--36.2

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

    8. Applied associate-/r/36.2

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

    9. Simplified19.3

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

    if 5.890069572265506e-16 < b_2

    1. Initial program 28.8

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

    2. Taylor expanded around inf 9.4

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.8719024022082672 \cdot 10^{+146}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 5.524249207244028 \cdot 10^{-127}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 5.890069572265506 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{a \cdot c}{-a \cdot c} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))