Average Error: 33.2 → 8.6
Time: 19.4s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.213987834649125 \cdot 10^{+64}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -2.16299334697662 \cdot 10^{-148}:\\ \;\;\;\;\frac{1}{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right) \cdot \frac{a}{a \cdot c}}\\ \mathbf{elif}\;b_2 \le 2.559678284282607 \cdot 10^{+69}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{\frac{1}{2}}{\frac{b_2}{c}}\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.213987834649125 \cdot 10^{+64}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -2.16299334697662 \cdot 10^{-148}:\\
\;\;\;\;\frac{1}{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right) \cdot \frac{a}{a \cdot c}}\\

\mathbf{elif}\;b_2 \le 2.559678284282607 \cdot 10^{+69}:\\
\;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{\frac{1}{2}}{\frac{b_2}{c}}\right)\\

\end{array}
double f(double a, double b_2, double c) {
        double r523767 = b_2;
        double r523768 = -r523767;
        double r523769 = r523767 * r523767;
        double r523770 = a;
        double r523771 = c;
        double r523772 = r523770 * r523771;
        double r523773 = r523769 - r523772;
        double r523774 = sqrt(r523773);
        double r523775 = r523768 - r523774;
        double r523776 = r523775 / r523770;
        return r523776;
}
↓
double f(double a, double b_2, double c) {
        double r523777 = b_2;
        double r523778 = -3.213987834649125e+64;
        bool r523779 = r523777 <= r523778;
        double r523780 = -0.5;
        double r523781 = c;
        double r523782 = r523781 / r523777;
        double r523783 = r523780 * r523782;
        double r523784 = -2.16299334697662e-148;
        bool r523785 = r523777 <= r523784;
        double r523786 = 1.0;
        double r523787 = r523777 * r523777;
        double r523788 = a;
        double r523789 = r523788 * r523781;
        double r523790 = r523787 - r523789;
        double r523791 = sqrt(r523790);
        double r523792 = r523791 - r523777;
        double r523793 = r523788 / r523789;
        double r523794 = r523792 * r523793;
        double r523795 = r523786 / r523794;
        double r523796 = 2.559678284282607e+69;
        bool r523797 = r523777 <= r523796;
        double r523798 = -r523777;
        double r523799 = r523798 - r523791;
        double r523800 = r523788 / r523799;
        double r523801 = r523786 / r523800;
        double r523802 = -2.0;
        double r523803 = r523777 / r523788;
        double r523804 = 0.5;
        double r523805 = r523777 / r523781;
        double r523806 = r523804 / r523805;
        double r523807 = fma(r523802, r523803, r523806);
        double r523808 = r523797 ? r523801 : r523807;
        double r523809 = r523785 ? r523795 : r523808;
        double r523810 = r523779 ? r523783 : r523809;
        return r523810;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -3.213987834649125e+64

    1. Initial program 56.9

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

    2. Taylor expanded around -inf 3.6

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

    if -3.213987834649125e+64 < b_2 < -2.16299334697662e-148

    1. Initial program 37.0

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

    3. Applied flip--37.1

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

    4. Simplified15.0

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

    5. Simplified15.0

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

    7. Applied clear-num15.1

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

    8. Simplified14.5

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

    if -2.16299334697662e-148 < b_2 < 2.559678284282607e+69

    1. Initial program 11.1

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

    3. Applied clear-num11.2

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

    5. Applied add-sqr-sqrt11.2

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

    6. Applied associate-/l*11.2

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

    7. Simplified11.2

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

    if 2.559678284282607e+69 < b_2

    1. Initial program 38.9

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

    2. Taylor expanded around inf 4.7

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

    3. Simplified4.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{\frac{1}{2}}{\frac{b_2}{c}}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.213987834649125 \cdot 10^{+64}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -2.16299334697662 \cdot 10^{-148}:\\ \;\;\;\;\frac{1}{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right) \cdot \frac{a}{a \cdot c}}\\ \mathbf{elif}\;b_2 \le 2.559678284282607 \cdot 10^{+69}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{\frac{1}{2}}{\frac{b_2}{c}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))