Average Error: 33.6 → 9.9
Time: 20.5s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.396811349079212 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(-2, \left(\frac{b_2}{a}\right), \left(\frac{1}{2} \cdot \frac{c}{b_2}\right)\right)\\ \mathbf{elif}\;b_2 \le 1.3659668388152999 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2} \cdot \frac{\sqrt{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.396811349079212 \cdot 10^{+61}:\\
\;\;\;\;\mathsf{fma}\left(-2, \left(\frac{b_2}{a}\right), \left(\frac{1}{2} \cdot \frac{c}{b_2}\right)\right)\\

\mathbf{elif}\;b_2 \le 1.3659668388152999 \cdot 10^{-67}:\\
\;\;\;\;\sqrt{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2} \cdot \frac{\sqrt{\sqrt{b_2 \cdot b_2 - c \cdot a} - 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 r704032 = b_2;
        double r704033 = -r704032;
        double r704034 = r704032 * r704032;
        double r704035 = a;
        double r704036 = c;
        double r704037 = r704035 * r704036;
        double r704038 = r704034 - r704037;
        double r704039 = sqrt(r704038);
        double r704040 = r704033 + r704039;
        double r704041 = r704040 / r704035;
        return r704041;
}
↓
double f(double a, double b_2, double c) {
        double r704042 = b_2;
        double r704043 = -3.396811349079212e+61;
        bool r704044 = r704042 <= r704043;
        double r704045 = -2.0;
        double r704046 = a;
        double r704047 = r704042 / r704046;
        double r704048 = 0.5;
        double r704049 = c;
        double r704050 = r704049 / r704042;
        double r704051 = r704048 * r704050;
        double r704052 = fma(r704045, r704047, r704051);
        double r704053 = 1.3659668388152999e-67;
        bool r704054 = r704042 <= r704053;
        double r704055 = r704042 * r704042;
        double r704056 = r704049 * r704046;
        double r704057 = r704055 - r704056;
        double r704058 = sqrt(r704057);
        double r704059 = r704058 - r704042;
        double r704060 = sqrt(r704059);
        double r704061 = r704060 / r704046;
        double r704062 = r704060 * r704061;
        double r704063 = -0.5;
        double r704064 = r704063 * r704050;
        double r704065 = r704054 ? r704062 : r704064;
        double r704066 = r704044 ? r704052 : r704065;
        return r704066;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -3.396811349079212e+61

    1. Initial program 37.6

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

    2. Simplified37.6

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

    3. Taylor expanded around -inf 4.3

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

    4. Simplified4.3

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

    if -3.396811349079212e+61 < b_2 < 1.3659668388152999e-67

    1. Initial program 13.8

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

    2. Simplified13.8

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

    4. Applied *-un-lft-identity13.8

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

    5. Applied add-sqr-sqrt14.2

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

    6. Applied times-frac14.2

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

    7. Simplified14.2

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

    if 1.3659668388152999e-67 < b_2

    1. Initial program 53.0

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

    2. Simplified53.0

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

    3. Taylor expanded around inf 8.1

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.396811349079212 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(-2, \left(\frac{b_2}{a}\right), \left(\frac{1}{2} \cdot \frac{c}{b_2}\right)\right)\\ \mathbf{elif}\;b_2 \le 1.3659668388152999 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2} \cdot \frac{\sqrt{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))