Average Error: 34.1 → 10.0
Time: 25.2s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.32337788234514 \cdot 10^{-89}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 5.121074390969514 \cdot 10^{+149}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -4.32337788234514 \cdot 10^{-89}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 5.121074390969514 \cdot 10^{+149}:\\
\;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot 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 r852128 = b_2;
        double r852129 = -r852128;
        double r852130 = r852128 * r852128;
        double r852131 = a;
        double r852132 = c;
        double r852133 = r852131 * r852132;
        double r852134 = r852130 - r852133;
        double r852135 = sqrt(r852134);
        double r852136 = r852129 - r852135;
        double r852137 = r852136 / r852131;
        return r852137;
}
↓
double f(double a, double b_2, double c) {
        double r852138 = b_2;
        double r852139 = -4.32337788234514e-89;
        bool r852140 = r852138 <= r852139;
        double r852141 = -0.5;
        double r852142 = c;
        double r852143 = r852142 / r852138;
        double r852144 = r852141 * r852143;
        double r852145 = 5.121074390969514e+149;
        bool r852146 = r852138 <= r852145;
        double r852147 = 1.0;
        double r852148 = a;
        double r852149 = -r852138;
        double r852150 = r852138 * r852138;
        double r852151 = r852142 * r852148;
        double r852152 = r852150 - r852151;
        double r852153 = sqrt(r852152);
        double r852154 = r852149 - r852153;
        double r852155 = r852148 / r852154;
        double r852156 = r852147 / r852155;
        double r852157 = 0.5;
        double r852158 = r852143 * r852157;
        double r852159 = 2.0;
        double r852160 = r852138 / r852148;
        double r852161 = r852159 * r852160;
        double r852162 = r852158 - r852161;
        double r852163 = r852146 ? r852156 : r852162;
        double r852164 = r852140 ? r852144 : r852163;
        return r852164;
}

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 < -4.32337788234514e-89

    1. Initial program 52.3

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

    2. Taylor expanded around -inf 9.2

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

    if -4.32337788234514e-89 < b_2 < 5.121074390969514e+149

    1. Initial program 12.3

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

    3. Applied *-un-lft-identity12.3

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

    4. Applied associate-/l*12.4

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

    if 5.121074390969514e+149 < b_2

    1. Initial program 59.1

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

    2. Taylor expanded around inf 2.6

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.32337788234514 \cdot 10^{-89}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 5.121074390969514 \cdot 10^{+149}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))