Average Error: 33.9 → 11.3
Time: 15.7s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.869662346631121401645595393947635525169 \cdot 10^{101}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.869662346631121401645595393947635525169 \cdot 10^{101}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\
\;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\

\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 r53030 = b_2;
        double r53031 = -r53030;
        double r53032 = r53030 * r53030;
        double r53033 = a;
        double r53034 = c;
        double r53035 = r53033 * r53034;
        double r53036 = r53032 - r53035;
        double r53037 = sqrt(r53036);
        double r53038 = r53031 - r53037;
        double r53039 = r53038 / r53033;
        return r53039;
}
↓
double f(double a, double b_2, double c) {
        double r53040 = b_2;
        double r53041 = -1.8696623466311214e+101;
        bool r53042 = r53040 <= r53041;
        double r53043 = -0.5;
        double r53044 = c;
        double r53045 = r53044 / r53040;
        double r53046 = r53043 * r53045;
        double r53047 = 7.455592343308264e-170;
        bool r53048 = r53040 <= r53047;
        double r53049 = r53040 * r53040;
        double r53050 = a;
        double r53051 = r53050 * r53044;
        double r53052 = r53049 - r53051;
        double r53053 = sqrt(r53052);
        double r53054 = r53053 - r53040;
        double r53055 = r53044 / r53054;
        double r53056 = 0.5;
        double r53057 = r53056 * r53045;
        double r53058 = 2.0;
        double r53059 = r53040 / r53050;
        double r53060 = r53058 * r53059;
        double r53061 = r53057 - r53060;
        double r53062 = r53048 ? r53055 : r53061;
        double r53063 = r53042 ? r53046 : r53062;
        return r53063;
}

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 3 regimes

  2. if b_2 < -1.8696623466311214e+101

    1. Initial program 59.8

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

    2. Taylor expanded around -inf 2.5

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

    if -1.8696623466311214e+101 < b_2 < 7.455592343308264e-170

    1. Initial program 28.9

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

    3. Applied flip--29.1

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

    4. Simplified16.6

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

    5. Simplified16.6

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

    7. Applied *-un-lft-identity16.6

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

    8. Applied *-un-lft-identity16.6

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

    9. Applied times-frac16.6

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

    10. Simplified16.6

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

    11. Simplified11.1

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

    if 7.455592343308264e-170 < b_2

    1. Initial program 23.0

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

    2. Taylor expanded around inf 17.1

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

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.869662346631121401645595393947635525169 \cdot 10^{101}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))