Average Error: 34.0 → 6.5
Time: 4.6s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.5940112039867074 \cdot 10^{100}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -2.11899432108077868 \cdot 10^{-304}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{elif}\;b_2 \le 1.55900006758493949 \cdot 10^{64}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.5940112039867074 \cdot 10^{100}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -2.11899432108077868 \cdot 10^{-304}:\\
\;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\

\mathbf{elif}\;b_2 \le 1.55900006758493949 \cdot 10^{64}:\\
\;\;\;\;\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 r15852 = b_2;
        double r15853 = -r15852;
        double r15854 = r15852 * r15852;
        double r15855 = a;
        double r15856 = c;
        double r15857 = r15855 * r15856;
        double r15858 = r15854 - r15857;
        double r15859 = sqrt(r15858);
        double r15860 = r15853 + r15859;
        double r15861 = r15860 / r15855;
        return r15861;
}
↓
double f(double a, double b_2, double c) {
        double r15862 = b_2;
        double r15863 = -3.5940112039867074e+100;
        bool r15864 = r15862 <= r15863;
        double r15865 = 0.5;
        double r15866 = c;
        double r15867 = r15866 / r15862;
        double r15868 = r15865 * r15867;
        double r15869 = 2.0;
        double r15870 = a;
        double r15871 = r15862 / r15870;
        double r15872 = r15869 * r15871;
        double r15873 = r15868 - r15872;
        double r15874 = -2.1189943210807787e-304;
        bool r15875 = r15862 <= r15874;
        double r15876 = 1.0;
        double r15877 = r15862 * r15862;
        double r15878 = r15870 * r15866;
        double r15879 = r15877 - r15878;
        double r15880 = sqrt(r15879);
        double r15881 = r15880 - r15862;
        double r15882 = r15870 / r15881;
        double r15883 = r15876 / r15882;
        double r15884 = 1.5590000675849395e+64;
        bool r15885 = r15862 <= r15884;
        double r15886 = -r15862;
        double r15887 = r15886 - r15880;
        double r15888 = r15866 / r15887;
        double r15889 = -0.5;
        double r15890 = r15889 * r15867;
        double r15891 = r15885 ? r15888 : r15890;
        double r15892 = r15875 ? r15883 : r15891;
        double r15893 = r15864 ? r15873 : r15892;
        return r15893;
}

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 < -3.5940112039867074e+100

    1. Initial program 47.3

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

    2. Taylor expanded around -inf 3.8

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

    if -3.5940112039867074e+100 < b_2 < -2.1189943210807787e-304

    1. Initial program 8.8

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

    3. Applied clear-num9.0

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

    4. Simplified9.0

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

    if -2.1189943210807787e-304 < b_2 < 1.5590000675849395e+64

    1. Initial program 29.5

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

    3. Applied flip-+29.5

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

    4. Simplified15.7

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

    6. Applied clear-num15.9

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

    7. Simplified15.5

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

    9. Applied clear-num15.5

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

    10. Simplified9.1

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

    12. Applied associate-/r*8.8

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

    13. Simplified8.7

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

    if 1.5590000675849395e+64 < b_2

    1. Initial program 57.3

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

    2. Taylor expanded around inf 3.5

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.5940112039867074 \cdot 10^{100}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -2.11899432108077868 \cdot 10^{-304}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{elif}\;b_2 \le 1.55900006758493949 \cdot 10^{64}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))