Average Error: 34.1 → 8.5
Time: 4.5s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.98705289364567391 \cdot 10^{52}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -4.35718478447219164 \cdot 10^{-283}:\\ \;\;\;\;\frac{\frac{1}{1}}{\frac{a}{a \cdot c} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}\\ \mathbf{elif}\;b_2 \le 7.9196559434345 \cdot 10^{101}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.98705289364567391 \cdot 10^{52}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -4.35718478447219164 \cdot 10^{-283}:\\
\;\;\;\;\frac{\frac{1}{1}}{\frac{a}{a \cdot c} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}\\

\mathbf{elif}\;b_2 \le 7.9196559434345 \cdot 10^{101}:\\
\;\;\;\;\frac{\left(-b_2\right) - \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 r88951 = b_2;
        double r88952 = -r88951;
        double r88953 = r88951 * r88951;
        double r88954 = a;
        double r88955 = c;
        double r88956 = r88954 * r88955;
        double r88957 = r88953 - r88956;
        double r88958 = sqrt(r88957);
        double r88959 = r88952 - r88958;
        double r88960 = r88959 / r88954;
        return r88960;
}
↓
double f(double a, double b_2, double c) {
        double r88961 = b_2;
        double r88962 = -3.987052893645674e+52;
        bool r88963 = r88961 <= r88962;
        double r88964 = -0.5;
        double r88965 = c;
        double r88966 = r88965 / r88961;
        double r88967 = r88964 * r88966;
        double r88968 = -4.357184784472192e-283;
        bool r88969 = r88961 <= r88968;
        double r88970 = 1.0;
        double r88971 = r88970 / r88970;
        double r88972 = a;
        double r88973 = r88972 * r88965;
        double r88974 = r88972 / r88973;
        double r88975 = r88961 * r88961;
        double r88976 = r88975 - r88973;
        double r88977 = sqrt(r88976);
        double r88978 = r88977 - r88961;
        double r88979 = r88974 * r88978;
        double r88980 = r88971 / r88979;
        double r88981 = 7.9196559434345e+101;
        bool r88982 = r88961 <= r88981;
        double r88983 = -r88961;
        double r88984 = r88983 - r88977;
        double r88985 = r88984 / r88972;
        double r88986 = 0.5;
        double r88987 = r88986 * r88966;
        double r88988 = 2.0;
        double r88989 = r88961 / r88972;
        double r88990 = r88988 * r88989;
        double r88991 = r88987 - r88990;
        double r88992 = r88982 ? r88985 : r88991;
        double r88993 = r88969 ? r88980 : r88992;
        double r88994 = r88963 ? r88967 : r88993;
        return r88994;
}

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 < -3.987052893645674e+52

    1. Initial program 57.7

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

    2. Taylor expanded around -inf 3.4

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

    if -3.987052893645674e+52 < b_2 < -4.357184784472192e-283

    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. Simplified16.4

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

    5. Simplified16.4

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

    7. Applied *-un-lft-identity16.4

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

    8. Applied *-un-lft-identity16.4

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

    9. Applied times-frac16.4

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

    10. Applied associate-/l*16.4

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

    11. Simplified16.2

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

    if -4.357184784472192e-283 < b_2 < 7.9196559434345e+101

    1. Initial program 9.5

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

    if 7.9196559434345e+101 < b_2

    1. Initial program 47.6

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

    2. Taylor expanded around inf 3.4

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

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.98705289364567391 \cdot 10^{52}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -4.35718478447219164 \cdot 10^{-283}:\\ \;\;\;\;\frac{\frac{1}{1}}{\frac{a}{a \cdot c} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}\\ \mathbf{elif}\;b_2 \le 7.9196559434345 \cdot 10^{101}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))