Average Error: 33.6 → 8.7
Time: 19.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.0756890097990247 \cdot 10^{+152}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.7117535086124408 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{c \cdot a}{a}}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\ \mathbf{elif}\;b_2 \le 2.8498539378971727 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -4.0756890097990247 \cdot 10^{+152}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -3.7117535086124408 \cdot 10^{-115}:\\
\;\;\;\;\frac{\frac{c \cdot a}{a}}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\

\mathbf{elif}\;b_2 \le 2.8498539378971727 \cdot 10^{+99}:\\
\;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r1716735 = b_2;
        double r1716736 = -r1716735;
        double r1716737 = r1716735 * r1716735;
        double r1716738 = a;
        double r1716739 = c;
        double r1716740 = r1716738 * r1716739;
        double r1716741 = r1716737 - r1716740;
        double r1716742 = sqrt(r1716741);
        double r1716743 = r1716736 - r1716742;
        double r1716744 = r1716743 / r1716738;
        return r1716744;
}
↓
double f(double a, double b_2, double c) {
        double r1716745 = b_2;
        double r1716746 = -4.0756890097990247e+152;
        bool r1716747 = r1716745 <= r1716746;
        double r1716748 = -0.5;
        double r1716749 = c;
        double r1716750 = r1716749 / r1716745;
        double r1716751 = r1716748 * r1716750;
        double r1716752 = -3.7117535086124408e-115;
        bool r1716753 = r1716745 <= r1716752;
        double r1716754 = a;
        double r1716755 = r1716749 * r1716754;
        double r1716756 = r1716755 / r1716754;
        double r1716757 = r1716745 * r1716745;
        double r1716758 = r1716757 - r1716755;
        double r1716759 = sqrt(r1716758);
        double r1716760 = r1716759 - r1716745;
        double r1716761 = r1716756 / r1716760;
        double r1716762 = 2.8498539378971727e+99;
        bool r1716763 = r1716745 <= r1716762;
        double r1716764 = 1.0;
        double r1716765 = r1716764 / r1716754;
        double r1716766 = -r1716745;
        double r1716767 = r1716766 - r1716759;
        double r1716768 = r1716765 * r1716767;
        double r1716769 = 0.5;
        double r1716770 = r1716750 * r1716769;
        double r1716771 = 2.0;
        double r1716772 = r1716745 / r1716754;
        double r1716773 = r1716771 * r1716772;
        double r1716774 = r1716770 - r1716773;
        double r1716775 = r1716763 ? r1716768 : r1716774;
        double r1716776 = r1716753 ? r1716761 : r1716775;
        double r1716777 = r1716747 ? r1716751 : r1716776;
        return r1716777;
}

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 < -4.0756890097990247e+152

    1. Initial program 62.7

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

    2. Taylor expanded around -inf 1.6

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

    if -4.0756890097990247e+152 < b_2 < -3.7117535086124408e-115

    1. Initial program 42.6

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

    3. Applied flip--42.7

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

    4. Simplified14.7

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

    5. Simplified14.7

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

    7. Applied *-un-lft-identity14.7

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

    8. Applied *-un-lft-identity14.7

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

    9. Applied *-un-lft-identity14.7

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

    10. Applied times-frac14.7

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

    11. Applied times-frac14.7

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

    12. Simplified14.7

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

    13. Simplified12.0

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

    if -3.7117535086124408e-115 < b_2 < 2.8498539378971727e+99

    1. Initial program 11.4

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

    3. Applied div-inv11.6

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

    if 2.8498539378971727e+99 < b_2

    1. Initial program 44.0

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

    2. Taylor expanded around inf 3.6

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

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.0756890097990247 \cdot 10^{+152}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.7117535086124408 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{c \cdot a}{a}}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\ \mathbf{elif}\;b_2 \le 2.8498539378971727 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))