Average Error: 34.1 → 8.5
Time: 4.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.98705289364567391 \cdot 10^{52}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -4.35718478447219164 \cdot 10^{-283}:\\ \;\;\;\;\frac{1}{\frac{a}{a \cdot c} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}\\ \mathbf{elif}\;b_2 \le 7.9196559434345 \cdot 10^{101}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\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{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 r14941 = b_2;
        double r14942 = -r14941;
        double r14943 = r14941 * r14941;
        double r14944 = a;
        double r14945 = c;
        double r14946 = r14944 * r14945;
        double r14947 = r14943 - r14946;
        double r14948 = sqrt(r14947);
        double r14949 = r14942 - r14948;
        double r14950 = r14949 / r14944;
        return r14950;
}
↓
double f(double a, double b_2, double c) {
        double r14951 = b_2;
        double r14952 = -3.987052893645674e+52;
        bool r14953 = r14951 <= r14952;
        double r14954 = -0.5;
        double r14955 = c;
        double r14956 = r14955 / r14951;
        double r14957 = r14954 * r14956;
        double r14958 = -4.357184784472192e-283;
        bool r14959 = r14951 <= r14958;
        double r14960 = 1.0;
        double r14961 = a;
        double r14962 = r14961 * r14955;
        double r14963 = r14961 / r14962;
        double r14964 = r14951 * r14951;
        double r14965 = r14964 - r14962;
        double r14966 = sqrt(r14965);
        double r14967 = r14966 - r14951;
        double r14968 = r14963 * r14967;
        double r14969 = r14960 / r14968;
        double r14970 = 7.9196559434345e+101;
        bool r14971 = r14951 <= r14970;
        double r14972 = -r14951;
        double r14973 = r14972 - r14966;
        double r14974 = r14973 / r14961;
        double r14975 = 0.5;
        double r14976 = r14975 * r14956;
        double r14977 = 2.0;
        double r14978 = r14951 / r14961;
        double r14979 = r14977 * r14978;
        double r14980 = r14976 - r14979;
        double r14981 = r14971 ? r14974 : r14980;
        double r14982 = r14959 ? r14969 : r14981;
        double r14983 = r14953 ? r14957 : r14982;
        return r14983;
}

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 clear-num16.4

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

    8. Simplified16.2

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

    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

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

Reproduce

herbie shell --seed 2020024 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))