Average Error: 19.6 → 13.1
Time: 22.5s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 2.891777552454845 \cdot 10^{+74}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)}} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;2 \cdot \frac{c}{2 \cdot \left(\frac{a}{b} \cdot c - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)} - b}{2 \cdot a}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 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 2.891777552454845 \cdot 10^{+74}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;2 \cdot \frac{c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)}}\\

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

\end{array}\\

\mathbf{elif}\;b \ge 0:\\
\;\;\;\;2 \cdot \frac{c}{2 \cdot \left(\frac{a}{b} \cdot c - b\right)}\\

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

\end{array}
double f(double a, double b, double c) {
        double r697589 = b;
        double r697590 = 0.0;
        bool r697591 = r697589 >= r697590;
        double r697592 = 2.0;
        double r697593 = c;
        double r697594 = r697592 * r697593;
        double r697595 = -r697589;
        double r697596 = r697589 * r697589;
        double r697597 = 4.0;
        double r697598 = a;
        double r697599 = r697597 * r697598;
        double r697600 = r697599 * r697593;
        double r697601 = r697596 - r697600;
        double r697602 = sqrt(r697601);
        double r697603 = r697595 - r697602;
        double r697604 = r697594 / r697603;
        double r697605 = r697595 + r697602;
        double r697606 = r697592 * r697598;
        double r697607 = r697605 / r697606;
        double r697608 = r697591 ? r697604 : r697607;
        return r697608;
}
↓
double f(double a, double b, double c) {
        double r697609 = b;
        double r697610 = 2.891777552454845e+74;
        bool r697611 = r697609 <= r697610;
        double r697612 = 0.0;
        bool r697613 = r697609 >= r697612;
        double r697614 = 2.0;
        double r697615 = c;
        double r697616 = -r697609;
        double r697617 = -4.0;
        double r697618 = a;
        double r697619 = r697618 * r697615;
        double r697620 = r697609 * r697609;
        double r697621 = fma(r697617, r697619, r697620);
        double r697622 = sqrt(r697621);
        double r697623 = r697616 - r697622;
        double r697624 = r697615 / r697623;
        double r697625 = r697614 * r697624;
        double r697626 = sqrt(r697622);
        double r697627 = r697626 * r697626;
        double r697628 = r697627 - r697609;
        double r697629 = r697614 * r697618;
        double r697630 = r697628 / r697629;
        double r697631 = r697613 ? r697625 : r697630;
        double r697632 = r697618 / r697609;
        double r697633 = r697632 * r697615;
        double r697634 = r697633 - r697609;
        double r697635 = r697614 * r697634;
        double r697636 = r697615 / r697635;
        double r697637 = r697614 * r697636;
        double r697638 = r697622 - r697609;
        double r697639 = r697638 / r697629;
        double r697640 = r697613 ? r697637 : r697639;
        double r697641 = r697611 ? r697631 : r697640;
        return r697641;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 2.891777552454845e+74

    1. Initial program 16.8

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    2. Simplified16.8

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, \left(c \cdot a\right), \left(b \cdot b\right)\right)}} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, \left(c \cdot a\right), \left(b \cdot b\right)\right)} - b}{2 \cdot a}\\ \end{array}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt16.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, \left(c \cdot a\right), \left(b \cdot b\right)\right)}} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{\mathsf{fma}\left(-4, \left(c \cdot a\right), \left(b \cdot b\right)\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, \left(c \cdot a\right), \left(b \cdot b\right)\right)}} - b}{2 \cdot a}\\ \end{array}\]

    if 2.891777552454845e+74 < b

    1. Initial program 27.2

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    2. Simplified27.2

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, \left(c \cdot a\right), \left(b \cdot b\right)\right)}} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, \left(c \cdot a\right), \left(b \cdot b\right)\right)} - b}{2 \cdot a}\\ \end{array}}\]

    3. Taylor expanded around inf 6.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, \left(c \cdot a\right), \left(b \cdot b\right)\right)} - b}{2 \cdot a}\\ \end{array}\]

    4. Simplified2.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c}{\color{blue}{\left(\frac{a}{b} \cdot c - b\right) \cdot 2}} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, \left(c \cdot a\right), \left(b \cdot b\right)\right)} - b}{2 \cdot a}\\ \end{array}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification13.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2.891777552454845 \cdot 10^{+74}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)}} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;2 \cdot \frac{c}{2 \cdot \left(\frac{a}{b} \cdot c - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)} - b}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 2"
  (if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))))