Average Error: 34.2 → 6.8
Time: 4.4s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -8.626773201174524 \cdot 10^{102}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -4.42774749682144966 \cdot 10^{-220}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{\frac{1}{c} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{elif}\;b_2 \le 1.6275304582996679 \cdot 10^{99}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -8.626773201174524 \cdot 10^{102}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -4.42774749682144966 \cdot 10^{-220}:\\
\;\;\;\;\frac{\frac{1}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{\frac{1}{c} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\

\mathbf{elif}\;b_2 \le 1.6275304582996679 \cdot 10^{99}:\\
\;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r80062 = b_2;
        double r80063 = -r80062;
        double r80064 = r80062 * r80062;
        double r80065 = a;
        double r80066 = c;
        double r80067 = r80065 * r80066;
        double r80068 = r80064 - r80067;
        double r80069 = sqrt(r80068);
        double r80070 = r80063 - r80069;
        double r80071 = r80070 / r80065;
        return r80071;
}
↓
double f(double a, double b_2, double c) {
        double r80072 = b_2;
        double r80073 = -8.626773201174524e+102;
        bool r80074 = r80072 <= r80073;
        double r80075 = -0.5;
        double r80076 = c;
        double r80077 = r80076 / r80072;
        double r80078 = r80075 * r80077;
        double r80079 = -4.42774749682145e-220;
        bool r80080 = r80072 <= r80079;
        double r80081 = 1.0;
        double r80082 = r80072 * r80072;
        double r80083 = a;
        double r80084 = r80083 * r80076;
        double r80085 = r80082 - r80084;
        double r80086 = sqrt(r80085);
        double r80087 = r80086 - r80072;
        double r80088 = sqrt(r80087);
        double r80089 = r80081 / r80088;
        double r80090 = r80081 / r80076;
        double r80091 = r80090 * r80088;
        double r80092 = r80089 / r80091;
        double r80093 = 1.627530458299668e+99;
        bool r80094 = r80072 <= r80093;
        double r80095 = -r80072;
        double r80096 = r80095 / r80083;
        double r80097 = r80086 / r80083;
        double r80098 = r80096 - r80097;
        double r80099 = 0.5;
        double r80100 = r80099 * r80077;
        double r80101 = 2.0;
        double r80102 = r80072 / r80083;
        double r80103 = r80101 * r80102;
        double r80104 = r80100 - r80103;
        double r80105 = r80094 ? r80098 : r80104;
        double r80106 = r80080 ? r80092 : r80105;
        double r80107 = r80074 ? r80078 : r80106;
        return r80107;
}

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 < -8.626773201174524e+102

    1. Initial program 59.4

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

    2. Taylor expanded around -inf 2.4

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

    if -8.626773201174524e+102 < b_2 < -4.42774749682145e-220

    1. Initial program 36.2

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

    3. Applied flip--36.3

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

    4. Simplified16.7

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

    5. Simplified16.7

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied add-sqr-sqrt16.9

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}}{a}\]

    8. Applied *-un-lft-identity16.9

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

    9. Applied times-frac16.9

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

    10. Applied associate-/l*16.3

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

    11. Simplified15.7

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

    13. Applied clear-num15.6

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

    14. Simplified7.7

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

    if -4.42774749682145e-220 < b_2 < 1.627530458299668e+99

    1. Initial program 10.3

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

    3. Applied div-sub10.4

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

    if 1.627530458299668e+99 < b_2

    1. Initial program 46.8

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

    2. Taylor expanded around inf 3.7

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -8.626773201174524 \cdot 10^{102}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -4.42774749682144966 \cdot 10^{-220}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{\frac{1}{c} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{elif}\;b_2 \le 1.6275304582996679 \cdot 10^{99}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))