Average Error: 34.3 → 6.9
Time: 7.0s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.9551520595513616 \cdot 10^{118}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -6.79526900931122647 \cdot 10^{-245}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{elif}\;b_2 \le 5.34931179548294658 \cdot 10^{30}:\\ \;\;\;\;\frac{\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{\frac{1}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.9551520595513616 \cdot 10^{118}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -6.79526900931122647 \cdot 10^{-245}:\\
\;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\

\mathbf{elif}\;b_2 \le 5.34931179548294658 \cdot 10^{30}:\\
\;\;\;\;\frac{\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{\frac{1}{c}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r25955 = b_2;
        double r25956 = -r25955;
        double r25957 = r25955 * r25955;
        double r25958 = a;
        double r25959 = c;
        double r25960 = r25958 * r25959;
        double r25961 = r25957 - r25960;
        double r25962 = sqrt(r25961);
        double r25963 = r25956 + r25962;
        double r25964 = r25963 / r25958;
        return r25964;
}
↓
double f(double a, double b_2, double c) {
        double r25965 = b_2;
        double r25966 = -5.955152059551362e+118;
        bool r25967 = r25965 <= r25966;
        double r25968 = 0.5;
        double r25969 = c;
        double r25970 = r25969 / r25965;
        double r25971 = r25968 * r25970;
        double r25972 = 2.0;
        double r25973 = a;
        double r25974 = r25965 / r25973;
        double r25975 = r25972 * r25974;
        double r25976 = r25971 - r25975;
        double r25977 = -6.7952690093112265e-245;
        bool r25978 = r25965 <= r25977;
        double r25979 = 1.0;
        double r25980 = r25965 * r25965;
        double r25981 = r25973 * r25969;
        double r25982 = r25980 - r25981;
        double r25983 = sqrt(r25982);
        double r25984 = r25983 - r25965;
        double r25985 = r25973 / r25984;
        double r25986 = r25979 / r25985;
        double r25987 = 5.349311795482947e+30;
        bool r25988 = r25965 <= r25987;
        double r25989 = -r25965;
        double r25990 = r25989 - r25983;
        double r25991 = r25979 / r25990;
        double r25992 = r25979 / r25969;
        double r25993 = r25991 / r25992;
        double r25994 = -0.5;
        double r25995 = r25994 * r25970;
        double r25996 = r25988 ? r25993 : r25995;
        double r25997 = r25978 ? r25986 : r25996;
        double r25998 = r25967 ? r25976 : r25997;
        return r25998;
}

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 < -5.955152059551362e+118

    1. Initial program 52.1

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

    2. Taylor expanded around -inf 2.9

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

    if -5.955152059551362e+118 < b_2 < -6.7952690093112265e-245

    1. Initial program 7.5

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

    3. Applied clear-num7.7

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

    4. Simplified7.7

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

    if -6.7952690093112265e-245 < b_2 < 5.349311795482947e+30

    1. Initial program 26.2

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

    3. Applied flip-+26.3

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

    4. Simplified16.5

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

    6. Applied *-un-lft-identity16.5

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

    7. Applied associate-/r*16.5

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

    8. Simplified14.3

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

    10. Applied div-inv14.4

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

    11. Applied *-un-lft-identity14.4

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

    12. Applied times-frac16.6

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

    13. Applied associate-/l*16.5

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

    14. Simplified10.5

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

    if 5.349311795482947e+30 < b_2

    1. Initial program 56.7

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

    2. Taylor expanded around inf 4.8

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

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.9551520595513616 \cdot 10^{118}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -6.79526900931122647 \cdot 10^{-245}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{elif}\;b_2 \le 5.34931179548294658 \cdot 10^{30}:\\ \;\;\;\;\frac{\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{\frac{1}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))