Average Error: 34.1 → 9.2
Time: 4.4s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.37749702272254886 \cdot 10^{101}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.9238883452280037 \cdot 10^{-130}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b_2 \le 4.01993084419163312 \cdot 10^{109}:\\ \;\;\;\;\frac{\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.37749702272254886 \cdot 10^{101}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 1.9238883452280037 \cdot 10^{-130}:\\
\;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\mathbf{elif}\;b_2 \le 4.01993084419163312 \cdot 10^{109}:\\
\;\;\;\;\frac{\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r14159 = b_2;
        double r14160 = -r14159;
        double r14161 = r14159 * r14159;
        double r14162 = a;
        double r14163 = c;
        double r14164 = r14162 * r14163;
        double r14165 = r14161 - r14164;
        double r14166 = sqrt(r14165);
        double r14167 = r14160 + r14166;
        double r14168 = r14167 / r14162;
        return r14168;
}
↓
double f(double a, double b_2, double c) {
        double r14169 = b_2;
        double r14170 = -2.377497022722549e+101;
        bool r14171 = r14169 <= r14170;
        double r14172 = 0.5;
        double r14173 = c;
        double r14174 = r14173 / r14169;
        double r14175 = r14172 * r14174;
        double r14176 = 2.0;
        double r14177 = a;
        double r14178 = r14169 / r14177;
        double r14179 = r14176 * r14178;
        double r14180 = r14175 - r14179;
        double r14181 = 1.9238883452280037e-130;
        bool r14182 = r14169 <= r14181;
        double r14183 = -r14169;
        double r14184 = r14169 * r14169;
        double r14185 = r14177 * r14173;
        double r14186 = r14184 - r14185;
        double r14187 = sqrt(r14186);
        double r14188 = r14183 + r14187;
        double r14189 = r14188 / r14177;
        double r14190 = 4.019930844191633e+109;
        bool r14191 = r14169 <= r14190;
        double r14192 = 0.0;
        double r14193 = r14192 + r14185;
        double r14194 = r14183 - r14187;
        double r14195 = r14193 / r14194;
        double r14196 = r14195 / r14177;
        double r14197 = -0.5;
        double r14198 = r14197 * r14174;
        double r14199 = r14191 ? r14196 : r14198;
        double r14200 = r14182 ? r14189 : r14199;
        double r14201 = r14171 ? r14180 : r14200;
        return r14201;
}

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 < -2.377497022722549e+101

    1. Initial program 47.1

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

    2. Taylor expanded around -inf 3.6

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

    if -2.377497022722549e+101 < b_2 < 1.9238883452280037e-130

    1. Initial program 11.9

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

    if 1.9238883452280037e-130 < b_2 < 4.019930844191633e+109

    1. Initial program 40.3

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

    3. Applied flip-+40.3

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

    4. Simplified15.5

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

    if 4.019930844191633e+109 < b_2

    1. Initial program 59.9

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

    2. Taylor expanded around inf 2.4

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

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.37749702272254886 \cdot 10^{101}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.9238883452280037 \cdot 10^{-130}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b_2 \le 4.01993084419163312 \cdot 10^{109}:\\ \;\;\;\;\frac{\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))