Average Error: 34.2 → 9.8
Time: 6.9s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.05249693088806959 \cdot 10^{141}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b_2 \le 5.08374808794434102 \cdot 10^{-70}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}, \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}, -b_2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.05249693088806959 \cdot 10^{141}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2}{a} \cdot -2\right)\\

\mathbf{elif}\;b_2 \le 5.08374808794434102 \cdot 10^{-70}:\\
\;\;\;\;\frac{\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}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r13766 = b_2;
        double r13767 = -r13766;
        double r13768 = r13766 * r13766;
        double r13769 = a;
        double r13770 = c;
        double r13771 = r13769 * r13770;
        double r13772 = r13768 - r13771;
        double r13773 = sqrt(r13772);
        double r13774 = r13767 + r13773;
        double r13775 = r13774 / r13769;
        return r13775;
}
↓
double f(double a, double b_2, double c) {
        double r13776 = b_2;
        double r13777 = -1.0524969308880696e+141;
        bool r13778 = r13776 <= r13777;
        double r13779 = 0.5;
        double r13780 = c;
        double r13781 = r13780 / r13776;
        double r13782 = a;
        double r13783 = r13776 / r13782;
        double r13784 = -2.0;
        double r13785 = r13783 * r13784;
        double r13786 = fma(r13779, r13781, r13785);
        double r13787 = 5.083748087944341e-70;
        bool r13788 = r13776 <= r13787;
        double r13789 = r13776 * r13776;
        double r13790 = r13782 * r13780;
        double r13791 = r13789 - r13790;
        double r13792 = sqrt(r13791);
        double r13793 = sqrt(r13792);
        double r13794 = -r13776;
        double r13795 = fma(r13793, r13793, r13794);
        double r13796 = r13795 / r13782;
        double r13797 = -0.5;
        double r13798 = r13797 * r13781;
        double r13799 = r13788 ? r13796 : r13798;
        double r13800 = r13778 ? r13786 : r13799;
        return r13800;
}

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 < -1.0524969308880696e+141

    1. Initial program 58.5

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

    2. Simplified58.5

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

    3. Taylor expanded around -inf 2.4

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

    4. Simplified2.4

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

    if -1.0524969308880696e+141 < b_2 < 5.083748087944341e-70

    1. Initial program 12.5

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

    2. Simplified12.5

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

    4. Applied add-sqr-sqrt12.5

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

    5. Applied sqrt-prod12.7

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

    6. Applied fma-neg12.7

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

    if 5.083748087944341e-70 < b_2

    1. Initial program 53.1

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

    2. Simplified53.1

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

    3. Taylor expanded around inf 8.6

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.05249693088806959 \cdot 10^{141}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b_2 \le 5.08374808794434102 \cdot 10^{-70}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}, \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}, -b_2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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))