Average Error: 33.7 → 10.9
Time: 11.4s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.4317863954914854 \cdot 10^{141}:\\ \;\;\;\;\left(\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.1860189201379418 \cdot 10^{-161}:\\ \;\;\;\;\frac{1}{a} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.4317863954914854 \cdot 10^{141}:\\
\;\;\;\;\left(\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 1.1860189201379418 \cdot 10^{-161}:\\
\;\;\;\;\frac{1}{a} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c} - \frac{b_2}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r14813 = b_2;
        double r14814 = -r14813;
        double r14815 = r14813 * r14813;
        double r14816 = a;
        double r14817 = c;
        double r14818 = r14816 * r14817;
        double r14819 = r14815 - r14818;
        double r14820 = sqrt(r14819);
        double r14821 = r14814 + r14820;
        double r14822 = r14821 / r14816;
        return r14822;
}
↓
double f(double a, double b_2, double c) {
        double r14823 = b_2;
        double r14824 = -2.4317863954914854e+141;
        bool r14825 = r14823 <= r14824;
        double r14826 = 0.5;
        double r14827 = c;
        double r14828 = r14827 / r14823;
        double r14829 = r14826 * r14828;
        double r14830 = a;
        double r14831 = r14823 / r14830;
        double r14832 = r14829 - r14831;
        double r14833 = r14832 - r14831;
        double r14834 = 1.1860189201379418e-161;
        bool r14835 = r14823 <= r14834;
        double r14836 = 1.0;
        double r14837 = r14836 / r14830;
        double r14838 = r14823 * r14823;
        double r14839 = r14830 * r14827;
        double r14840 = r14838 - r14839;
        double r14841 = sqrt(r14840);
        double r14842 = r14837 * r14841;
        double r14843 = r14842 - r14831;
        double r14844 = -0.5;
        double r14845 = r14844 * r14828;
        double r14846 = r14835 ? r14843 : r14845;
        double r14847 = r14825 ? r14833 : r14846;
        return r14847;
}

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 < -2.4317863954914854e+141

    1. Initial program 59.4

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

    2. Simplified59.4

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
    3. Using strategy rm

    4. Applied div-sub59.4

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

    5. Taylor expanded around -inf 2.9

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

    if -2.4317863954914854e+141 < b_2 < 1.1860189201379418e-161

    1. Initial program 10.2

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

    2. Simplified10.2

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
    3. Using strategy rm

    4. Applied div-sub10.2

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity10.2

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

    7. Applied sqrt-prod10.2

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

    8. Applied associate-/l*10.3

      \[\leadsto \color{blue}{\frac{\sqrt{1}}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c}}}} - \frac{b_2}{a}\]
    9. Using strategy rm

    10. Applied div-inv10.4

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

    11. Applied add-cube-cbrt10.4

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{1}} \cdot \sqrt[3]{\sqrt{1}}\right) \cdot \sqrt[3]{\sqrt{1}}}}{a \cdot \frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c}}} - \frac{b_2}{a}\]

    12. Applied times-frac10.4

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\sqrt{1}} \cdot \sqrt[3]{\sqrt{1}}}{a} \cdot \frac{\sqrt[3]{\sqrt{1}}}{\frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c}}}} - \frac{b_2}{a}\]

    13. Simplified10.4

      \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{\sqrt[3]{\sqrt{1}}}{\frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c}}} - \frac{b_2}{a}\]

    14. Simplified10.3

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

    if 1.1860189201379418e-161 < b_2

    1. Initial program 49.6

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

    2. Simplified49.6

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

    3. Taylor expanded around inf 13.6

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

  4. Final simplification10.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.4317863954914854 \cdot 10^{141}:\\ \;\;\;\;\left(\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.1860189201379418 \cdot 10^{-161}:\\ \;\;\;\;\frac{1}{a} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))