Average Error: 19.8 → 8.2
Time: 20.2s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.177741376353205990919022742706386584896 \cdot 10^{137}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{a \cdot c}{b}, -2 \cdot b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 8.991400549328153943320663038409734113411 \cdot 10^{83}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{a \cdot c}{b}, -2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.177741376353205990919022742706386584896 \cdot 10^{137}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

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

\end{array}\\

\mathbf{elif}\;b \le 8.991400549328153943320663038409734113411 \cdot 10^{83}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\

\end{array}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{a \cdot c}{b}, -2 \cdot b\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\end{array}
double f(double a, double b, double c) {
        double r1118773 = b;
        double r1118774 = 0.0;
        bool r1118775 = r1118773 >= r1118774;
        double r1118776 = 2.0;
        double r1118777 = c;
        double r1118778 = r1118776 * r1118777;
        double r1118779 = -r1118773;
        double r1118780 = r1118773 * r1118773;
        double r1118781 = 4.0;
        double r1118782 = a;
        double r1118783 = r1118781 * r1118782;
        double r1118784 = r1118783 * r1118777;
        double r1118785 = r1118780 - r1118784;
        double r1118786 = sqrt(r1118785);
        double r1118787 = r1118779 - r1118786;
        double r1118788 = r1118778 / r1118787;
        double r1118789 = r1118779 + r1118786;
        double r1118790 = r1118776 * r1118782;
        double r1118791 = r1118789 / r1118790;
        double r1118792 = r1118775 ? r1118788 : r1118791;
        return r1118792;
}
↓
double f(double a, double b, double c) {
        double r1118793 = b;
        double r1118794 = -1.177741376353206e+137;
        bool r1118795 = r1118793 <= r1118794;
        double r1118796 = 0.0;
        bool r1118797 = r1118793 >= r1118796;
        double r1118798 = 2.0;
        double r1118799 = c;
        double r1118800 = r1118798 * r1118799;
        double r1118801 = -r1118793;
        double r1118802 = r1118793 * r1118793;
        double r1118803 = 4.0;
        double r1118804 = a;
        double r1118805 = r1118803 * r1118804;
        double r1118806 = r1118805 * r1118799;
        double r1118807 = r1118802 - r1118806;
        double r1118808 = sqrt(r1118807);
        double r1118809 = r1118801 - r1118808;
        double r1118810 = r1118800 / r1118809;
        double r1118811 = r1118804 * r1118799;
        double r1118812 = r1118811 / r1118793;
        double r1118813 = -2.0;
        double r1118814 = r1118813 * r1118793;
        double r1118815 = fma(r1118798, r1118812, r1118814);
        double r1118816 = r1118798 * r1118804;
        double r1118817 = r1118815 / r1118816;
        double r1118818 = r1118797 ? r1118810 : r1118817;
        double r1118819 = 8.991400549328154e+83;
        bool r1118820 = r1118793 <= r1118819;
        double r1118821 = sqrt(r1118808);
        double r1118822 = r1118821 * r1118821;
        double r1118823 = r1118801 + r1118822;
        double r1118824 = r1118823 / r1118816;
        double r1118825 = r1118797 ? r1118810 : r1118824;
        double r1118826 = r1118800 / r1118815;
        double r1118827 = r1118801 + r1118808;
        double r1118828 = r1118827 / r1118816;
        double r1118829 = r1118797 ? r1118826 : r1118828;
        double r1118830 = r1118820 ? r1118825 : r1118829;
        double r1118831 = r1118795 ? r1118818 : r1118830;
        return r1118831;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -1.177741376353206e+137

    1. Initial program 57.9

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    2. Taylor expanded around -inf 9.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2 \cdot a}\\ \end{array}\]

    3. Simplified9.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{a \cdot c}{b}, -2 \cdot b\right)}{2 \cdot a}\\ \end{array}\]

    if -1.177741376353206e+137 < b < 8.991400549328154e+83

    1. Initial program 8.4

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt8.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]

    4. Applied sqrt-prod8.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]

    if 8.991400549328154e+83 < b

    1. Initial program 28.2

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    2. Taylor expanded around inf 6.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    3. Simplified6.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\mathsf{fma}\left(2, \frac{a \cdot c}{b}, -2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.177741376353205990919022742706386584896 \cdot 10^{137}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{a \cdot c}{b}, -2 \cdot b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 8.991400549328153943320663038409734113411 \cdot 10^{83}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{a \cdot c}{b}, -2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 2"
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))