Average Error: 34.1 → 9.2
Time: 4.5s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.37749702272254886 \cdot 10^{101}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.9238883452280037 \cdot 10^{-130}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b_2 \le 4.01993084419163312 \cdot 10^{109}:\\ \;\;\;\;\frac{\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\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 r14103 = b_2;
        double r14104 = -r14103;
        double r14105 = r14103 * r14103;
        double r14106 = a;
        double r14107 = c;
        double r14108 = r14106 * r14107;
        double r14109 = r14105 - r14108;
        double r14110 = sqrt(r14109);
        double r14111 = r14104 + r14110;
        double r14112 = r14111 / r14106;
        return r14112;
}
↓
double f(double a, double b_2, double c) {
        double r14113 = b_2;
        double r14114 = -2.377497022722549e+101;
        bool r14115 = r14113 <= r14114;
        double r14116 = 0.5;
        double r14117 = c;
        double r14118 = r14117 / r14113;
        double r14119 = r14116 * r14118;
        double r14120 = 2.0;
        double r14121 = a;
        double r14122 = r14113 / r14121;
        double r14123 = r14120 * r14122;
        double r14124 = r14119 - r14123;
        double r14125 = 1.9238883452280037e-130;
        bool r14126 = r14113 <= r14125;
        double r14127 = -r14113;
        double r14128 = r14113 * r14113;
        double r14129 = r14121 * r14117;
        double r14130 = r14128 - r14129;
        double r14131 = sqrt(r14130);
        double r14132 = r14127 + r14131;
        double r14133 = r14132 / r14121;
        double r14134 = 4.019930844191633e+109;
        bool r14135 = r14113 <= r14134;
        double r14136 = 0.0;
        double r14137 = r14136 + r14129;
        double r14138 = r14127 - r14131;
        double r14139 = r14137 / r14138;
        double r14140 = r14139 / r14121;
        double r14141 = -0.5;
        double r14142 = r14141 * r14118;
        double r14143 = r14135 ? r14140 : r14142;
        double r14144 = r14126 ? r14133 : r14143;
        double r14145 = r14115 ? r14124 : r14144;
        return r14145;
}

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))