Average Error: 34.2 → 6.8
Time: 4.7s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -8.626773201174524 \cdot 10^{102}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -4.42774749682144966 \cdot 10^{-220}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{\frac{1}{c} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{elif}\;b_2 \le 1.6275304582996679 \cdot 10^{99}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -8.626773201174524 \cdot 10^{102}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -4.42774749682144966 \cdot 10^{-220}:\\
\;\;\;\;\frac{\frac{1}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{\frac{1}{c} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\

\mathbf{elif}\;b_2 \le 1.6275304582996679 \cdot 10^{99}:\\
\;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r14746 = b_2;
        double r14747 = -r14746;
        double r14748 = r14746 * r14746;
        double r14749 = a;
        double r14750 = c;
        double r14751 = r14749 * r14750;
        double r14752 = r14748 - r14751;
        double r14753 = sqrt(r14752);
        double r14754 = r14747 - r14753;
        double r14755 = r14754 / r14749;
        return r14755;
}
↓
double f(double a, double b_2, double c) {
        double r14756 = b_2;
        double r14757 = -8.626773201174524e+102;
        bool r14758 = r14756 <= r14757;
        double r14759 = -0.5;
        double r14760 = c;
        double r14761 = r14760 / r14756;
        double r14762 = r14759 * r14761;
        double r14763 = -4.42774749682145e-220;
        bool r14764 = r14756 <= r14763;
        double r14765 = 1.0;
        double r14766 = r14756 * r14756;
        double r14767 = a;
        double r14768 = r14767 * r14760;
        double r14769 = r14766 - r14768;
        double r14770 = sqrt(r14769);
        double r14771 = r14770 - r14756;
        double r14772 = sqrt(r14771);
        double r14773 = r14765 / r14772;
        double r14774 = r14765 / r14760;
        double r14775 = r14774 * r14772;
        double r14776 = r14773 / r14775;
        double r14777 = 1.627530458299668e+99;
        bool r14778 = r14756 <= r14777;
        double r14779 = -r14756;
        double r14780 = r14779 / r14767;
        double r14781 = r14770 / r14767;
        double r14782 = r14780 - r14781;
        double r14783 = 0.5;
        double r14784 = r14783 * r14761;
        double r14785 = 2.0;
        double r14786 = r14756 / r14767;
        double r14787 = r14785 * r14786;
        double r14788 = r14784 - r14787;
        double r14789 = r14778 ? r14782 : r14788;
        double r14790 = r14764 ? r14776 : r14789;
        double r14791 = r14758 ? r14762 : r14790;
        return r14791;
}

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 < -8.626773201174524e+102

    1. Initial program 59.4

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

    2. Taylor expanded around -inf 2.4

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

    if -8.626773201174524e+102 < b_2 < -4.42774749682145e-220

    1. Initial program 36.2

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

    3. Applied flip--36.3

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

    4. Simplified16.7

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

    5. Simplified16.7

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied add-sqr-sqrt16.9

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}}{a}\]

    8. Applied *-un-lft-identity16.9

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

    9. Applied times-frac16.9

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

    10. Applied associate-/l*16.3

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

    11. Simplified15.7

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

    13. Applied clear-num15.6

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

    14. Simplified7.7

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

    if -4.42774749682145e-220 < b_2 < 1.627530458299668e+99

    1. Initial program 10.3

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

    3. Applied div-sub10.4

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

    if 1.627530458299668e+99 < b_2

    1. Initial program 46.8

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

    2. Taylor expanded around inf 3.7

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -8.626773201174524 \cdot 10^{102}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -4.42774749682144966 \cdot 10^{-220}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{\frac{1}{c} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{elif}\;b_2 \le 1.6275304582996679 \cdot 10^{99}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))