Average Error: 34.4 → 8.3
Time: 6.2s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.9644058459680186 \cdot 10^{71}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -8.61562639771237576 \cdot 10^{-228}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 1.2011474830795371 \cdot 10^{35}:\\ \;\;\;\;\frac{1 \cdot \frac{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{\frac{1}{c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.9644058459680186 \cdot 10^{71}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -8.61562639771237576 \cdot 10^{-228}:\\
\;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

\mathbf{elif}\;b_2 \le 1.2011474830795371 \cdot 10^{35}:\\
\;\;\;\;\frac{1 \cdot \frac{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{\frac{1}{c}}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r20798 = b_2;
        double r20799 = -r20798;
        double r20800 = r20798 * r20798;
        double r20801 = a;
        double r20802 = c;
        double r20803 = r20801 * r20802;
        double r20804 = r20800 - r20803;
        double r20805 = sqrt(r20804);
        double r20806 = r20799 + r20805;
        double r20807 = r20806 / r20801;
        return r20807;
}
↓
double f(double a, double b_2, double c) {
        double r20808 = b_2;
        double r20809 = -2.9644058459680186e+71;
        bool r20810 = r20808 <= r20809;
        double r20811 = 0.5;
        double r20812 = c;
        double r20813 = r20812 / r20808;
        double r20814 = r20811 * r20813;
        double r20815 = 2.0;
        double r20816 = a;
        double r20817 = r20808 / r20816;
        double r20818 = r20815 * r20817;
        double r20819 = r20814 - r20818;
        double r20820 = -8.615626397712376e-228;
        bool r20821 = r20808 <= r20820;
        double r20822 = -r20808;
        double r20823 = r20808 * r20808;
        double r20824 = r20816 * r20812;
        double r20825 = r20823 - r20824;
        double r20826 = sqrt(r20825);
        double r20827 = r20822 + r20826;
        double r20828 = 1.0;
        double r20829 = r20828 / r20816;
        double r20830 = r20827 * r20829;
        double r20831 = 1.2011474830795371e+35;
        bool r20832 = r20808 <= r20831;
        double r20833 = r20822 - r20826;
        double r20834 = r20816 / r20833;
        double r20835 = r20828 / r20812;
        double r20836 = r20834 / r20835;
        double r20837 = r20828 * r20836;
        double r20838 = r20837 / r20816;
        double r20839 = -0.5;
        double r20840 = r20839 * r20813;
        double r20841 = r20832 ? r20838 : r20840;
        double r20842 = r20821 ? r20830 : r20841;
        double r20843 = r20810 ? r20819 : r20842;
        return r20843;
}

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 < -2.9644058459680186e+71

    1. Initial program 42.3

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

    2. Taylor expanded around -inf 4.6

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

    if -2.9644058459680186e+71 < b_2 < -8.615626397712376e-228

    1. Initial program 9.2

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

    3. Applied div-inv9.4

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

    if -8.615626397712376e-228 < b_2 < 1.2011474830795371e+35

    1. Initial program 26.0

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

    3. Applied flip-+26.2

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

    4. Simplified16.7

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

    6. Applied *-un-lft-identity16.7

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

    7. Applied *-un-lft-identity16.7

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

    8. Applied times-frac16.7

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

    9. Simplified16.7

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

    10. Simplified13.8

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

    12. Applied div-inv13.8

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

    13. Applied associate-/r*13.9

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

    if 1.2011474830795371e+35 < b_2

    1. Initial program 57.1

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

    2. Taylor expanded around inf 4.0

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

  4. Final simplification8.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.9644058459680186 \cdot 10^{71}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -8.61562639771237576 \cdot 10^{-228}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 1.2011474830795371 \cdot 10^{35}:\\ \;\;\;\;\frac{1 \cdot \frac{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{\frac{1}{c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))