Average Error: 34.1 → 12.5
Time: 6.0s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.5812949048043538 \cdot 10^{-96}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 7.99853652545151242 \cdot 10^{49}:\\ \;\;\;\;\frac{\frac{1}{1}}{\frac{a}{a \cdot c} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.5812949048043538 \cdot 10^{-96}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 7.99853652545151242 \cdot 10^{49}:\\
\;\;\;\;\frac{\frac{1}{1}}{\frac{a}{a \cdot c} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r19774 = b_2;
        double r19775 = -r19774;
        double r19776 = r19774 * r19774;
        double r19777 = a;
        double r19778 = c;
        double r19779 = r19777 * r19778;
        double r19780 = r19776 - r19779;
        double r19781 = sqrt(r19780);
        double r19782 = r19775 + r19781;
        double r19783 = r19782 / r19777;
        return r19783;
}
↓
double f(double a, double b_2, double c) {
        double r19784 = b_2;
        double r19785 = -3.581294904804354e-96;
        bool r19786 = r19784 <= r19785;
        double r19787 = 0.5;
        double r19788 = c;
        double r19789 = r19788 / r19784;
        double r19790 = r19787 * r19789;
        double r19791 = 2.0;
        double r19792 = a;
        double r19793 = r19784 / r19792;
        double r19794 = r19791 * r19793;
        double r19795 = r19790 - r19794;
        double r19796 = 7.998536525451512e+49;
        bool r19797 = r19784 <= r19796;
        double r19798 = 1.0;
        double r19799 = r19798 / r19798;
        double r19800 = r19792 * r19788;
        double r19801 = r19792 / r19800;
        double r19802 = -r19784;
        double r19803 = r19784 * r19784;
        double r19804 = r19803 - r19800;
        double r19805 = sqrt(r19804);
        double r19806 = r19802 - r19805;
        double r19807 = r19801 * r19806;
        double r19808 = r19799 / r19807;
        double r19809 = -0.5;
        double r19810 = r19809 * r19789;
        double r19811 = r19797 ? r19808 : r19810;
        double r19812 = r19786 ? r19795 : r19811;
        return r19812;
}

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 < -3.581294904804354e-96

    1. Initial program 25.9

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

    2. Taylor expanded around -inf 12.9

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

    if -3.581294904804354e-96 < b_2 < 7.998536525451512e+49

    1. Initial program 24.5

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

    3. Applied flip-+26.7

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

    4. Simplified18.5

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

    6. Applied *-un-lft-identity18.5

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

    7. Applied *-un-lft-identity18.5

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

    8. Applied times-frac18.5

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

    9. Applied associate-/l*18.6

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

    10. Simplified18.4

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

    if 7.998536525451512e+49 < b_2

    1. Initial program 56.9

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

    2. Taylor expanded around inf 4.0

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

  4. Final simplification12.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.5812949048043538 \cdot 10^{-96}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 7.99853652545151242 \cdot 10^{49}:\\ \;\;\;\;\frac{\frac{1}{1}}{\frac{a}{a \cdot c} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 +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))