Average Error: 34.1 → 7.4
Time: 6.8s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.8371925747446876 \cdot 10^{53}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -4.0623007329414777 \cdot 10^{-248}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 1.77017414835012383 \cdot 10^{70}:\\ \;\;\;\;\frac{1}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -4.8371925747446876 \cdot 10^{53}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -4.0623007329414777 \cdot 10^{-248}:\\
\;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

\mathbf{elif}\;b_2 \le 1.77017414835012383 \cdot 10^{70}:\\
\;\;\;\;\frac{1}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r24785 = b_2;
        double r24786 = -r24785;
        double r24787 = r24785 * r24785;
        double r24788 = a;
        double r24789 = c;
        double r24790 = r24788 * r24789;
        double r24791 = r24787 - r24790;
        double r24792 = sqrt(r24791);
        double r24793 = r24786 + r24792;
        double r24794 = r24793 / r24788;
        return r24794;
}
↓
double f(double a, double b_2, double c) {
        double r24795 = b_2;
        double r24796 = -4.837192574744688e+53;
        bool r24797 = r24795 <= r24796;
        double r24798 = 0.5;
        double r24799 = c;
        double r24800 = r24799 / r24795;
        double r24801 = r24798 * r24800;
        double r24802 = 2.0;
        double r24803 = a;
        double r24804 = r24795 / r24803;
        double r24805 = r24802 * r24804;
        double r24806 = r24801 - r24805;
        double r24807 = -4.062300732941478e-248;
        bool r24808 = r24795 <= r24807;
        double r24809 = -r24795;
        double r24810 = r24795 * r24795;
        double r24811 = r24803 * r24799;
        double r24812 = r24810 - r24811;
        double r24813 = sqrt(r24812);
        double r24814 = r24809 + r24813;
        double r24815 = 1.0;
        double r24816 = r24815 / r24803;
        double r24817 = r24814 * r24816;
        double r24818 = 1.7701741483501238e+70;
        bool r24819 = r24795 <= r24818;
        double r24820 = r24809 - r24813;
        double r24821 = r24820 / r24799;
        double r24822 = r24815 / r24821;
        double r24823 = -0.5;
        double r24824 = r24823 * r24800;
        double r24825 = r24819 ? r24822 : r24824;
        double r24826 = r24808 ? r24817 : r24825;
        double r24827 = r24797 ? r24806 : r24826;
        return r24827;
}

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 4 regimes

  2. if b_2 < -4.837192574744688e+53

    1. Initial program 37.6

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

    2. Taylor expanded around -inf 5.6

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

    if -4.837192574744688e+53 < b_2 < -4.062300732941478e-248

    1. Initial program 9.2

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied div-inv9.4

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

    if -4.062300732941478e-248 < b_2 < 1.7701741483501238e+70

    1. Initial program 28.7

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied flip-+28.7

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

    4. Simplified16.0

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity16.0

      \[\leadsto \frac{\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{\color{blue}{1 \cdot a}}\]

    7. Applied associate-/r*16.0

      \[\leadsto \color{blue}{\frac{\frac{\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{1}}{a}}\]

    8. Simplified14.0

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

    10. Applied *-un-lft-identity14.0

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

    11. Applied *-un-lft-identity14.0

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

    12. Applied times-frac14.0

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

    13. Applied associate-/l*14.0

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

    14. Simplified10.0

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

    if 1.7701741483501238e+70 < b_2

    1. Initial program 58.5

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

    2. Taylor expanded around inf 3.8

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification7.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.8371925747446876 \cdot 10^{53}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -4.0623007329414777 \cdot 10^{-248}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 1.77017414835012383 \cdot 10^{70}:\\ \;\;\;\;\frac{1}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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))