Average Error: 33.6 → 10.6
Time: 18.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -7.363255598823911 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.823572975982288 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{b_2 \cdot b_2 - \left(b_2 \cdot b_2 - c \cdot a\right)}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - c \cdot a}}\\ \mathbf{elif}\;b_2 \le -2.3344326820285623 \cdot 10^{-123}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.6691257204922504 \cdot 10^{+85}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{1}{2} \cdot \frac{c}{b_2}\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -7.363255598823911 \cdot 10^{-15}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -1.823572975982288 \cdot 10^{-27}:\\
\;\;\;\;\frac{\frac{b_2 \cdot b_2 - \left(b_2 \cdot b_2 - c \cdot a\right)}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - c \cdot a}}\\

\mathbf{elif}\;b_2 \le -2.3344326820285623 \cdot 10^{-123}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.6691257204922504 \cdot 10^{+85}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r2709757 = b_2;
        double r2709758 = -r2709757;
        double r2709759 = r2709757 * r2709757;
        double r2709760 = a;
        double r2709761 = c;
        double r2709762 = r2709760 * r2709761;
        double r2709763 = r2709759 - r2709762;
        double r2709764 = sqrt(r2709763);
        double r2709765 = r2709758 - r2709764;
        double r2709766 = r2709765 / r2709760;
        return r2709766;
}
↓
double f(double a, double b_2, double c) {
        double r2709767 = b_2;
        double r2709768 = -7.363255598823911e-15;
        bool r2709769 = r2709767 <= r2709768;
        double r2709770 = -0.5;
        double r2709771 = c;
        double r2709772 = r2709771 / r2709767;
        double r2709773 = r2709770 * r2709772;
        double r2709774 = -1.823572975982288e-27;
        bool r2709775 = r2709767 <= r2709774;
        double r2709776 = r2709767 * r2709767;
        double r2709777 = a;
        double r2709778 = r2709771 * r2709777;
        double r2709779 = r2709776 - r2709778;
        double r2709780 = r2709776 - r2709779;
        double r2709781 = r2709780 / r2709777;
        double r2709782 = -r2709767;
        double r2709783 = sqrt(r2709779);
        double r2709784 = r2709782 + r2709783;
        double r2709785 = r2709781 / r2709784;
        double r2709786 = -2.3344326820285623e-123;
        bool r2709787 = r2709767 <= r2709786;
        double r2709788 = 1.6691257204922504e+85;
        bool r2709789 = r2709767 <= r2709788;
        double r2709790 = r2709782 - r2709783;
        double r2709791 = r2709790 / r2709777;
        double r2709792 = -2.0;
        double r2709793 = r2709767 / r2709777;
        double r2709794 = 0.5;
        double r2709795 = r2709794 * r2709772;
        double r2709796 = fma(r2709792, r2709793, r2709795);
        double r2709797 = r2709789 ? r2709791 : r2709796;
        double r2709798 = r2709787 ? r2709773 : r2709797;
        double r2709799 = r2709775 ? r2709785 : r2709798;
        double r2709800 = r2709769 ? r2709773 : r2709799;
        return r2709800;
}

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 < -7.363255598823911e-15 or -1.823572975982288e-27 < b_2 < -2.3344326820285623e-123

    1. Initial program 50.8

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

    2. Taylor expanded around -inf 10.6

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

    if -7.363255598823911e-15 < b_2 < -1.823572975982288e-27

    1. Initial program 36.1

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

    3. Applied clear-num36.1

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

    5. Applied flip--36.2

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

    6. Applied associate-/r/36.2

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

    7. Applied associate-/r*36.2

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

    8. Simplified36.1

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

    if -2.3344326820285623e-123 < b_2 < 1.6691257204922504e+85

    1. Initial program 12.6

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

    3. Applied div-inv12.7

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

    5. Applied un-div-inv12.6

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

    if 1.6691257204922504e+85 < b_2

    1. Initial program 42.9

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

    3. Applied div-inv42.9

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

    4. Taylor expanded around inf 3.6

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

    5. Simplified3.6

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -7.363255598823911 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.823572975982288 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{b_2 \cdot b_2 - \left(b_2 \cdot b_2 - c \cdot a\right)}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - c \cdot a}}\\ \mathbf{elif}\;b_2 \le -2.3344326820285623 \cdot 10^{-123}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.6691257204922504 \cdot 10^{+85}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{1}{2} \cdot \frac{c}{b_2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))