Average Error: 33.6 → 10.4
Time: 22.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.1144981103869975 \cdot 10^{+131}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 4.5810084990875205 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)} - b}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.1144981103869975 \cdot 10^{+131}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 4.5810084990875205 \cdot 10^{-68}:\\
\;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)} - b}}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1550804 = b;
        double r1550805 = -r1550804;
        double r1550806 = r1550804 * r1550804;
        double r1550807 = 4.0;
        double r1550808 = a;
        double r1550809 = r1550807 * r1550808;
        double r1550810 = c;
        double r1550811 = r1550809 * r1550810;
        double r1550812 = r1550806 - r1550811;
        double r1550813 = sqrt(r1550812);
        double r1550814 = r1550805 + r1550813;
        double r1550815 = 2.0;
        double r1550816 = r1550815 * r1550808;
        double r1550817 = r1550814 / r1550816;
        return r1550817;
}
↓
double f(double a, double b, double c) {
        double r1550818 = b;
        double r1550819 = -2.1144981103869975e+131;
        bool r1550820 = r1550818 <= r1550819;
        double r1550821 = c;
        double r1550822 = r1550821 / r1550818;
        double r1550823 = a;
        double r1550824 = r1550818 / r1550823;
        double r1550825 = r1550822 - r1550824;
        double r1550826 = 2.0;
        double r1550827 = r1550825 * r1550826;
        double r1550828 = r1550827 / r1550826;
        double r1550829 = 4.5810084990875205e-68;
        bool r1550830 = r1550818 <= r1550829;
        double r1550831 = 1.0;
        double r1550832 = -4.0;
        double r1550833 = r1550832 * r1550823;
        double r1550834 = r1550821 * r1550833;
        double r1550835 = fma(r1550818, r1550818, r1550834);
        double r1550836 = sqrt(r1550835);
        double r1550837 = r1550836 - r1550818;
        double r1550838 = r1550823 / r1550837;
        double r1550839 = r1550831 / r1550838;
        double r1550840 = r1550839 / r1550826;
        double r1550841 = -2.0;
        double r1550842 = r1550841 * r1550822;
        double r1550843 = r1550842 / r1550826;
        double r1550844 = r1550830 ? r1550840 : r1550843;
        double r1550845 = r1550820 ? r1550828 : r1550844;
        return r1550845;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -2.1144981103869975e+131

    1. Initial program 53.8

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Simplified53.8

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}{a}}{2}}\]

    3. Taylor expanded around -inf 2.6

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

    4. Simplified2.6

      \[\leadsto \frac{\color{blue}{2 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}}{2}\]

    if -2.1144981103869975e+131 < b < 4.5810084990875205e-68

    1. Initial program 13.3

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Simplified13.3

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}{a}}{2}}\]
    3. Using strategy rm

    4. Applied clear-num13.5

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}}}}{2}\]

    if 4.5810084990875205e-68 < b

    1. Initial program 52.0

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Simplified52.0

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}{a}}{2}}\]

    3. Taylor expanded around inf 9.3

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.1144981103869975 \cdot 10^{+131}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 4.5810084990875205 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)} - b}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))