Average Error: 34.4 → 10.5
Time: 18.2s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.221067196710922123169723133116561516447 \cdot 10^{149}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{c}{b_2} \cdot \frac{1}{2}\right)\\ \mathbf{elif}\;b_2 \le 3.142311858008121469027865121070306475283 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.221067196710922123169723133116561516447 \cdot 10^{149}:\\
\;\;\;\;\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{c}{b_2} \cdot \frac{1}{2}\right)\\

\mathbf{elif}\;b_2 \le 3.142311858008121469027865121070306475283 \cdot 10^{-35}:\\
\;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r726635 = b_2;
        double r726636 = -r726635;
        double r726637 = r726635 * r726635;
        double r726638 = a;
        double r726639 = c;
        double r726640 = r726638 * r726639;
        double r726641 = r726637 - r726640;
        double r726642 = sqrt(r726641);
        double r726643 = r726636 + r726642;
        double r726644 = r726643 / r726638;
        return r726644;
}
↓
double f(double a, double b_2, double c) {
        double r726645 = b_2;
        double r726646 = -2.221067196710922e+149;
        bool r726647 = r726645 <= r726646;
        double r726648 = -2.0;
        double r726649 = a;
        double r726650 = r726645 / r726649;
        double r726651 = c;
        double r726652 = r726651 / r726645;
        double r726653 = 0.5;
        double r726654 = r726652 * r726653;
        double r726655 = fma(r726648, r726650, r726654);
        double r726656 = 3.1423118580081215e-35;
        bool r726657 = r726645 <= r726656;
        double r726658 = -r726645;
        double r726659 = r726645 * r726645;
        double r726660 = r726651 * r726649;
        double r726661 = r726659 - r726660;
        double r726662 = sqrt(r726661);
        double r726663 = r726658 + r726662;
        double r726664 = r726663 / r726649;
        double r726665 = -0.5;
        double r726666 = r726665 * r726652;
        double r726667 = r726657 ? r726664 : r726666;
        double r726668 = r726647 ? r726655 : r726667;
        return r726668;
}

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 < -2.221067196710922e+149

    1. Initial program 62.3

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

    2. Simplified62.3

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

    3. Taylor expanded around -inf 2.7

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

    4. Simplified2.7

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

    if -2.221067196710922e+149 < b_2 < 3.1423118580081215e-35

    1. Initial program 14.5

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

    2. Simplified14.5

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

    4. Applied add-sqr-sqrt14.5

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

    5. Applied sqrt-prod14.7

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

    6. Applied fma-neg14.7

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}, \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}, -b_2\right)}}{a}\]
    7. Using strategy rm

    8. Applied fma-udef14.7

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

    9. Simplified14.5

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

    if 3.1423118580081215e-35 < b_2

    1. Initial program 54.4

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

    2. Simplified54.4

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

    3. Taylor expanded around inf 7.3

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.221067196710922123169723133116561516447 \cdot 10^{149}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{c}{b_2} \cdot \frac{1}{2}\right)\\ \mathbf{elif}\;b_2 \le 3.142311858008121469027865121070306475283 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +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))