Average Error: 34.2 → 9.8
Time: 7.0s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.05249693088806959 \cdot 10^{141}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, -2 \cdot \frac{b_2}{a}\right)\\ \mathbf{elif}\;b_2 \le 5.08374808794434102 \cdot 10^{-70}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}, \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}, -b_2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.05249693088806959 \cdot 10^{141}:\\
\;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, -2 \cdot \frac{b_2}{a}\right)\\

\mathbf{elif}\;b_2 \le 5.08374808794434102 \cdot 10^{-70}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}, \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}, -b_2\right)}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r13832 = b_2;
        double r13833 = -r13832;
        double r13834 = r13832 * r13832;
        double r13835 = a;
        double r13836 = c;
        double r13837 = r13835 * r13836;
        double r13838 = r13834 - r13837;
        double r13839 = sqrt(r13838);
        double r13840 = r13833 + r13839;
        double r13841 = r13840 / r13835;
        return r13841;
}
↓
double f(double a, double b_2, double c) {
        double r13842 = b_2;
        double r13843 = -1.0524969308880696e+141;
        bool r13844 = r13842 <= r13843;
        double r13845 = c;
        double r13846 = r13845 / r13842;
        double r13847 = 0.5;
        double r13848 = -2.0;
        double r13849 = a;
        double r13850 = r13842 / r13849;
        double r13851 = r13848 * r13850;
        double r13852 = fma(r13846, r13847, r13851);
        double r13853 = 5.083748087944341e-70;
        bool r13854 = r13842 <= r13853;
        double r13855 = r13842 * r13842;
        double r13856 = r13849 * r13845;
        double r13857 = r13855 - r13856;
        double r13858 = sqrt(r13857);
        double r13859 = sqrt(r13858);
        double r13860 = -r13842;
        double r13861 = fma(r13859, r13859, r13860);
        double r13862 = r13861 / r13849;
        double r13863 = -0.5;
        double r13864 = r13863 * r13846;
        double r13865 = r13854 ? r13862 : r13864;
        double r13866 = r13844 ? r13852 : r13865;
        return r13866;
}

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 < -1.0524969308880696e+141

    1. Initial program 58.5

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

    2. Simplified58.5

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

    3. Taylor expanded around -inf 2.4

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

    4. Simplified2.4

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

    if -1.0524969308880696e+141 < b_2 < 5.083748087944341e-70

    1. Initial program 12.5

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

    2. Simplified12.5

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

    4. Applied add-sqr-sqrt12.5

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

    5. Applied sqrt-prod12.7

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

    6. Applied fma-neg12.7

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

    if 5.083748087944341e-70 < b_2

    1. Initial program 53.1

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

    2. Simplified53.1

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

    3. Taylor expanded around inf 8.6

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.05249693088806959 \cdot 10^{141}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, -2 \cdot \frac{b_2}{a}\right)\\ \mathbf{elif}\;b_2 \le 5.08374808794434102 \cdot 10^{-70}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}, \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}, -b_2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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))