Average Error: 33.0 → 10.6
Time: 18.7s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.794505329565205 \cdot 10^{+146}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 1.6194276288860963:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.794505329565205 \cdot 10^{+146}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\

\mathbf{elif}\;b_2 \le 1.6194276288860963:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r535780 = b_2;
        double r535781 = -r535780;
        double r535782 = r535780 * r535780;
        double r535783 = a;
        double r535784 = c;
        double r535785 = r535783 * r535784;
        double r535786 = r535782 - r535785;
        double r535787 = sqrt(r535786);
        double r535788 = r535781 + r535787;
        double r535789 = r535788 / r535783;
        return r535789;
}
↓
double f(double a, double b_2, double c) {
        double r535790 = b_2;
        double r535791 = -3.794505329565205e+146;
        bool r535792 = r535790 <= r535791;
        double r535793 = 0.5;
        double r535794 = c;
        double r535795 = r535794 / r535790;
        double r535796 = r535793 * r535795;
        double r535797 = a;
        double r535798 = r535790 / r535797;
        double r535799 = 2.0;
        double r535800 = r535798 * r535799;
        double r535801 = r535796 - r535800;
        double r535802 = 1.6194276288860963;
        bool r535803 = r535790 <= r535802;
        double r535804 = r535790 * r535790;
        double r535805 = r535794 * r535797;
        double r535806 = r535804 - r535805;
        double r535807 = sqrt(r535806);
        double r535808 = r535807 - r535790;
        double r535809 = r535808 / r535797;
        double r535810 = -0.5;
        double r535811 = r535810 * r535795;
        double r535812 = r535803 ? r535809 : r535811;
        double r535813 = r535792 ? r535801 : r535812;
        return r535813;
}

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 < -3.794505329565205e+146

    1. Initial program 58.0

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

    2. Simplified58.0

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

    3. Taylor expanded around -inf 3.0

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

    if -3.794505329565205e+146 < b_2 < 1.6194276288860963

    1. Initial program 15.0

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

    2. Simplified15.0

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

    4. Applied *-un-lft-identity15.0

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

    5. Applied associate-/l*15.1

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity15.1

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

    8. Applied *-un-lft-identity15.1

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

    9. Applied times-frac15.1

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

    10. Applied add-sqr-sqrt15.1

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

    11. Applied times-frac15.1

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

    12. Simplified15.1

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

    13. Simplified15.0

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

    if 1.6194276288860963 < b_2

    1. Initial program 54.4

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

    2. Simplified54.4

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

    3. Taylor expanded around inf 5.8

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.794505329565205 \cdot 10^{+146}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 1.6194276288860963:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))