Average Error: 32.7 → 10.3
Time: 20.9s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.3093766494757864 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 1.2352405779465016 \cdot 10^{-131}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.3093766494757864 \cdot 10^{+77}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\

\mathbf{elif}\;b_2 \le 1.2352405779465016 \cdot 10^{-131}:\\
\;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r589624 = b_2;
        double r589625 = -r589624;
        double r589626 = r589624 * r589624;
        double r589627 = a;
        double r589628 = c;
        double r589629 = r589627 * r589628;
        double r589630 = r589626 - r589629;
        double r589631 = sqrt(r589630);
        double r589632 = r589625 + r589631;
        double r589633 = r589632 / r589627;
        return r589633;
}
↓
double f(double a, double b_2, double c) {
        double r589634 = b_2;
        double r589635 = -2.3093766494757864e+77;
        bool r589636 = r589634 <= r589635;
        double r589637 = 0.5;
        double r589638 = c;
        double r589639 = r589638 / r589634;
        double r589640 = r589637 * r589639;
        double r589641 = a;
        double r589642 = r589634 / r589641;
        double r589643 = 2.0;
        double r589644 = r589642 * r589643;
        double r589645 = r589640 - r589644;
        double r589646 = 1.2352405779465016e-131;
        bool r589647 = r589634 <= r589646;
        double r589648 = 1.0;
        double r589649 = r589634 * r589634;
        double r589650 = r589638 * r589641;
        double r589651 = r589649 - r589650;
        double r589652 = sqrt(r589651);
        double r589653 = r589652 - r589634;
        double r589654 = r589641 / r589653;
        double r589655 = r589648 / r589654;
        double r589656 = -0.5;
        double r589657 = r589656 * r589639;
        double r589658 = r589647 ? r589655 : r589657;
        double r589659 = r589636 ? r589645 : r589658;
        return r589659;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -2.3093766494757864e+77

    1. Initial program 39.5

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

    2. Simplified39.5

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

    3. Taylor expanded around -inf 5.0

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

    if -2.3093766494757864e+77 < b_2 < 1.2352405779465016e-131

    1. Initial program 11.0

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

    2. Simplified11.0

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

    4. Applied *-un-lft-identity11.0

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

    5. Applied associate-/l*11.1

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

    if 1.2352405779465016e-131 < b_2

    1. Initial program 50.3

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

    2. Simplified50.3

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

    3. Taylor expanded around inf 11.7

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.3093766494757864 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 1.2352405779465016 \cdot 10^{-131}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 +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))