Average Error: 34.7 → 6.7
Time: 5.6s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.9721792334777768 \cdot 10^{151}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 7.48477236865457127 \cdot 10^{-258}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b_2 \le 9.79208634865372271 \cdot 10^{126}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.9721792334777768 \cdot 10^{151}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 7.48477236865457127 \cdot 10^{-258}:\\
\;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\mathbf{elif}\;b_2 \le 9.79208634865372271 \cdot 10^{126}:\\
\;\;\;\;\frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r17128 = b_2;
        double r17129 = -r17128;
        double r17130 = r17128 * r17128;
        double r17131 = a;
        double r17132 = c;
        double r17133 = r17131 * r17132;
        double r17134 = r17130 - r17133;
        double r17135 = sqrt(r17134);
        double r17136 = r17129 + r17135;
        double r17137 = r17136 / r17131;
        return r17137;
}
↓
double f(double a, double b_2, double c) {
        double r17138 = b_2;
        double r17139 = -1.9721792334777768e+151;
        bool r17140 = r17138 <= r17139;
        double r17141 = 0.5;
        double r17142 = c;
        double r17143 = r17142 / r17138;
        double r17144 = r17141 * r17143;
        double r17145 = 2.0;
        double r17146 = a;
        double r17147 = r17138 / r17146;
        double r17148 = r17145 * r17147;
        double r17149 = r17144 - r17148;
        double r17150 = 7.484772368654571e-258;
        bool r17151 = r17138 <= r17150;
        double r17152 = -r17138;
        double r17153 = r17138 * r17138;
        double r17154 = r17146 * r17142;
        double r17155 = r17153 - r17154;
        double r17156 = sqrt(r17155);
        double r17157 = r17152 + r17156;
        double r17158 = r17157 / r17146;
        double r17159 = 9.792086348653723e+126;
        bool r17160 = r17138 <= r17159;
        double r17161 = r17152 - r17156;
        double r17162 = r17142 / r17161;
        double r17163 = -0.5;
        double r17164 = r17163 * r17143;
        double r17165 = r17160 ? r17162 : r17164;
        double r17166 = r17151 ? r17158 : r17165;
        double r17167 = r17140 ? r17149 : r17166;
        return r17167;
}

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 < -1.9721792334777768e+151

    1. Initial program 62.9

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

    2. Taylor expanded around -inf 1.9

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

    if -1.9721792334777768e+151 < b_2 < 7.484772368654571e-258

    1. Initial program 9.8

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

    if 7.484772368654571e-258 < b_2 < 9.792086348653723e+126

    1. Initial program 34.4

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

    3. Applied flip-+34.5

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

    4. Simplified16.0

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

    6. Applied clear-num16.1

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

    7. Simplified14.8

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

    9. Applied associate-/r*14.6

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

    10. Simplified8.1

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

    if 9.792086348653723e+126 < b_2

    1. Initial program 61.4

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

    2. Taylor expanded around inf 2.1

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

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.9721792334777768 \cdot 10^{151}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 7.48477236865457127 \cdot 10^{-258}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b_2 \le 9.79208634865372271 \cdot 10^{126}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))