Average Error: 32.8 → 9.8
Time: 20.1s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.3599228730895225 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{1}{2} \cdot \frac{c}{b_2}\right)\\ \mathbf{elif}\;b_2 \le 3.1295384133612364 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.3599228730895225 \cdot 10^{+90}:\\
\;\;\;\;\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{1}{2} \cdot \frac{c}{b_2}\right)\\

\mathbf{elif}\;b_2 \le 3.1295384133612364 \cdot 10^{-73}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\

\end{array}
double f(double a, double b_2, double c) {
        double r352516 = b_2;
        double r352517 = -r352516;
        double r352518 = r352516 * r352516;
        double r352519 = a;
        double r352520 = c;
        double r352521 = r352519 * r352520;
        double r352522 = r352518 - r352521;
        double r352523 = sqrt(r352522);
        double r352524 = r352517 + r352523;
        double r352525 = r352524 / r352519;
        return r352525;
}
↓
double f(double a, double b_2, double c) {
        double r352526 = b_2;
        double r352527 = -1.3599228730895225e+90;
        bool r352528 = r352526 <= r352527;
        double r352529 = -2.0;
        double r352530 = a;
        double r352531 = r352526 / r352530;
        double r352532 = 0.5;
        double r352533 = c;
        double r352534 = r352533 / r352526;
        double r352535 = r352532 * r352534;
        double r352536 = fma(r352529, r352531, r352535);
        double r352537 = 3.1295384133612364e-73;
        bool r352538 = r352526 <= r352537;
        double r352539 = r352526 * r352526;
        double r352540 = r352533 * r352530;
        double r352541 = r352539 - r352540;
        double r352542 = sqrt(r352541);
        double r352543 = r352542 - r352526;
        double r352544 = r352543 / r352530;
        double r352545 = -0.5;
        double r352546 = r352534 * r352545;
        double r352547 = r352538 ? r352544 : r352546;
        double r352548 = r352528 ? r352536 : r352547;
        return r352548;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -1.3599228730895225e+90

    1. Initial program 41.6

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

    2. Simplified41.6

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

    3. Taylor expanded around -inf 3.8

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

    4. Simplified3.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{1}{2} \cdot \frac{c}{b_2}\right)}\]

    if -1.3599228730895225e+90 < b_2 < 3.1295384133612364e-73

    1. Initial program 12.8

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

    2. Simplified12.8

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

    3. Taylor expanded around inf 12.8

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

    4. Simplified12.8

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

    if 3.1295384133612364e-73 < b_2

    1. Initial program 52.3

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

    2. Simplified52.3

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

    3. Taylor expanded around inf 52.3

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

    4. Simplified52.3

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

    5. Taylor expanded around inf 9.0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.3599228730895225 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{1}{2} \cdot \frac{c}{b_2}\right)\\ \mathbf{elif}\;b_2 \le 3.1295384133612364 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))