Average Error: 18.9 → 13.1
Time: 22.0s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 4.1813913858897907 \cdot 10^{+68}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(-1, b, \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}} \cdot \left(-\sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}\right)\right) + \mathsf{fma}\left(-\sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]
\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 4.1813913858897907 \cdot 10^{+68}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(-1, b, \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}} \cdot \left(-\sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}\right)\right) + \mathsf{fma}\left(-\sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}\right)}\\

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

\end{array}\\

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

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

\end{array}
double f(double a, double b, double c) {
        double r812595 = b;
        double r812596 = 0.0;
        bool r812597 = r812595 >= r812596;
        double r812598 = 2.0;
        double r812599 = c;
        double r812600 = r812598 * r812599;
        double r812601 = -r812595;
        double r812602 = r812595 * r812595;
        double r812603 = 4.0;
        double r812604 = a;
        double r812605 = r812603 * r812604;
        double r812606 = r812605 * r812599;
        double r812607 = r812602 - r812606;
        double r812608 = sqrt(r812607);
        double r812609 = r812601 - r812608;
        double r812610 = r812600 / r812609;
        double r812611 = r812601 + r812608;
        double r812612 = r812598 * r812604;
        double r812613 = r812611 / r812612;
        double r812614 = r812597 ? r812610 : r812613;
        return r812614;
}
↓
double f(double a, double b, double c) {
        double r812615 = b;
        double r812616 = 4.1813913858897907e+68;
        bool r812617 = r812615 <= r812616;
        double r812618 = 0.0;
        bool r812619 = r812615 >= r812618;
        double r812620 = 2.0;
        double r812621 = c;
        double r812622 = r812620 * r812621;
        double r812623 = -1.0;
        double r812624 = -4.0;
        double r812625 = a;
        double r812626 = r812625 * r812621;
        double r812627 = r812615 * r812615;
        double r812628 = fma(r812624, r812626, r812627);
        double r812629 = sqrt(r812628);
        double r812630 = sqrt(r812629);
        double r812631 = -r812630;
        double r812632 = r812630 * r812631;
        double r812633 = fma(r812623, r812615, r812632);
        double r812634 = r812630 * r812630;
        double r812635 = fma(r812631, r812630, r812634);
        double r812636 = r812633 + r812635;
        double r812637 = r812622 / r812636;
        double r812638 = r812629 - r812615;
        double r812639 = r812638 / r812620;
        double r812640 = r812639 / r812625;
        double r812641 = r812619 ? r812637 : r812640;
        double r812642 = -r812615;
        double r812643 = r812642 - r812615;
        double r812644 = r812622 / r812643;
        double r812645 = r812619 ? r812644 : r812640;
        double r812646 = r812617 ? r812641 : r812645;
        return r812646;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 4.1813913858897907e+68

    1. Initial program 16.3

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

    2. Simplified16.3

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

    4. Applied add-sqr-sqrt16.3

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

    5. Applied sqrt-prod16.4

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

    6. Applied neg-mul-116.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{-1 \cdot b} - \sqrt{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]

    7. Applied prod-diff16.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\mathsf{fma}\left(-1, b, -\sqrt{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}\right) + \mathsf{fma}\left(-\sqrt{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]

    if 4.1813913858897907e+68 < b

    1. Initial program 26.4

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

    2. Simplified26.4

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

    3. Taylor expanded around 0 3.5

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

  4. Final simplification13.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 4.1813913858897907 \cdot 10^{+68}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(-1, b, \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}} \cdot \left(-\sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}\right)\right) + \mathsf{fma}\left(-\sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +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))))