Average Error: 34.6 → 10.6
Time: 1.2m
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.766818940874854722177248139872145176232 \cdot 10^{100}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b_2 \le 7.923524897992036987166355557663274472861 \cdot 10^{-153}:\\ \;\;\;\;\left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.766818940874854722177248139872145176232 \cdot 10^{100}:\\
\;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\

\mathbf{elif}\;b_2 \le 7.923524897992036987166355557663274472861 \cdot 10^{-153}:\\
\;\;\;\;\left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right) \cdot \frac{1}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r737771 = b_2;
        double r737772 = -r737771;
        double r737773 = r737771 * r737771;
        double r737774 = a;
        double r737775 = c;
        double r737776 = r737774 * r737775;
        double r737777 = r737773 - r737776;
        double r737778 = sqrt(r737777);
        double r737779 = r737772 + r737778;
        double r737780 = r737779 / r737774;
        return r737780;
}
↓
double f(double a, double b_2, double c) {
        double r737781 = b_2;
        double r737782 = -2.7668189408748547e+100;
        bool r737783 = r737781 <= r737782;
        double r737784 = c;
        double r737785 = r737784 / r737781;
        double r737786 = 0.5;
        double r737787 = a;
        double r737788 = r737781 / r737787;
        double r737789 = -2.0;
        double r737790 = r737788 * r737789;
        double r737791 = fma(r737785, r737786, r737790);
        double r737792 = 7.923524897992037e-153;
        bool r737793 = r737781 <= r737792;
        double r737794 = r737781 * r737781;
        double r737795 = r737787 * r737784;
        double r737796 = r737794 - r737795;
        double r737797 = sqrt(r737796);
        double r737798 = r737797 - r737781;
        double r737799 = 1.0;
        double r737800 = r737799 / r737787;
        double r737801 = r737798 * r737800;
        double r737802 = -0.5;
        double r737803 = r737802 * r737785;
        double r737804 = r737793 ? r737801 : r737803;
        double r737805 = r737783 ? r737791 : r737804;
        return r737805;
}

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 < -2.7668189408748547e+100

    1. Initial program 47.1

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

    2. Simplified47.1

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

    4. Applied div-inv47.2

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

    5. Taylor expanded around -inf 4.0

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

    6. Simplified4.0

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

    if -2.7668189408748547e+100 < b_2 < 7.923524897992037e-153

    1. Initial program 10.8

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

    2. Simplified10.8

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

    4. Applied div-inv10.9

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

    if 7.923524897992037e-153 < b_2

    1. Initial program 50.5

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

    2. Simplified50.5

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

    3. Taylor expanded around inf 12.7

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.766818940874854722177248139872145176232 \cdot 10^{100}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b_2 \le 7.923524897992036987166355557663274472861 \cdot 10^{-153}:\\ \;\;\;\;\left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +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))