Average Error: 34.1 → 9.8
Time: 21.6s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.008402374157600307221015587992064225208 \cdot 10^{154}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{-2 \cdot b_2}{a}\right)\\ \mathbf{elif}\;b_2 \le 1.61145084478121505718169973575148582501 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.008402374157600307221015587992064225208 \cdot 10^{154}:\\
\;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{-2 \cdot b_2}{a}\right)\\

\mathbf{elif}\;b_2 \le 1.61145084478121505718169973575148582501 \cdot 10^{-34}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r24566 = b_2;
        double r24567 = -r24566;
        double r24568 = r24566 * r24566;
        double r24569 = a;
        double r24570 = c;
        double r24571 = r24569 * r24570;
        double r24572 = r24568 - r24571;
        double r24573 = sqrt(r24572);
        double r24574 = r24567 + r24573;
        double r24575 = r24574 / r24569;
        return r24575;
}
↓
double f(double a, double b_2, double c) {
        double r24576 = b_2;
        double r24577 = -1.0084023741576003e+154;
        bool r24578 = r24576 <= r24577;
        double r24579 = c;
        double r24580 = r24579 / r24576;
        double r24581 = 0.5;
        double r24582 = -2.0;
        double r24583 = r24582 * r24576;
        double r24584 = a;
        double r24585 = r24583 / r24584;
        double r24586 = fma(r24580, r24581, r24585);
        double r24587 = 1.611450844781215e-34;
        bool r24588 = r24576 <= r24587;
        double r24589 = r24576 * r24576;
        double r24590 = r24584 * r24579;
        double r24591 = r24589 - r24590;
        double r24592 = sqrt(r24591);
        double r24593 = r24592 / r24584;
        double r24594 = r24576 / r24584;
        double r24595 = r24593 - r24594;
        double r24596 = -0.5;
        double r24597 = r24596 * r24580;
        double r24598 = r24588 ? r24595 : r24597;
        double r24599 = r24578 ? r24586 : r24598;
        return r24599;
}

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 < -1.0084023741576003e+154

    1. Initial program 64.0

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

    2. Simplified64.0

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
    3. Using strategy rm

    4. Applied div-inv64.0

      \[\leadsto \color{blue}{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right) \cdot \frac{1}{a}}\]

    5. Taylor expanded around -inf 1.7

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

    6. Simplified1.7

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

    if -1.0084023741576003e+154 < b_2 < 1.611450844781215e-34

    1. Initial program 13.5

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

    2. Simplified13.5

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
    3. Using strategy rm

    4. Applied div-sub13.5

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

    if 1.611450844781215e-34 < b_2

    1. Initial program 55.0

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

    2. Simplified55.0

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

    3. Taylor expanded around inf 7.0

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.008402374157600307221015587992064225208 \cdot 10^{154}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{-2 \cdot b_2}{a}\right)\\ \mathbf{elif}\;b_2 \le 1.61145084478121505718169973575148582501 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))