Average Error: 33.7 → 7.4
Time: 17.4s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -227369802444031.66:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -2.851839700712213 \cdot 10^{-304}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 4.71744724099961 \cdot 10^{+65}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -227369802444031.66:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -2.851839700712213 \cdot 10^{-304}:\\
\;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\

\mathbf{elif}\;b_2 \le 4.71744724099961 \cdot 10^{+65}:\\
\;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r1545633 = b_2;
        double r1545634 = -r1545633;
        double r1545635 = r1545633 * r1545633;
        double r1545636 = a;
        double r1545637 = c;
        double r1545638 = r1545636 * r1545637;
        double r1545639 = r1545635 - r1545638;
        double r1545640 = sqrt(r1545639);
        double r1545641 = r1545634 - r1545640;
        double r1545642 = r1545641 / r1545636;
        return r1545642;
}
↓
double f(double a, double b_2, double c) {
        double r1545643 = b_2;
        double r1545644 = -227369802444031.66;
        bool r1545645 = r1545643 <= r1545644;
        double r1545646 = -0.5;
        double r1545647 = c;
        double r1545648 = r1545647 / r1545643;
        double r1545649 = r1545646 * r1545648;
        double r1545650 = -2.851839700712213e-304;
        bool r1545651 = r1545643 <= r1545650;
        double r1545652 = r1545643 * r1545643;
        double r1545653 = a;
        double r1545654 = r1545653 * r1545647;
        double r1545655 = r1545652 - r1545654;
        double r1545656 = sqrt(r1545655);
        double r1545657 = r1545656 - r1545643;
        double r1545658 = r1545647 / r1545657;
        double r1545659 = 4.71744724099961e+65;
        bool r1545660 = r1545643 <= r1545659;
        double r1545661 = -r1545643;
        double r1545662 = r1545661 - r1545656;
        double r1545663 = 1.0;
        double r1545664 = r1545663 / r1545653;
        double r1545665 = r1545662 * r1545664;
        double r1545666 = -2.0;
        double r1545667 = r1545643 * r1545666;
        double r1545668 = r1545667 / r1545653;
        double r1545669 = r1545660 ? r1545665 : r1545668;
        double r1545670 = r1545651 ? r1545658 : r1545669;
        double r1545671 = r1545645 ? r1545649 : r1545670;
        return r1545671;
}

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 < -227369802444031.66

    1. Initial program 55.8

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

    2. Taylor expanded around -inf 5.0

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

    if -227369802444031.66 < b_2 < -2.851839700712213e-304

    1. Initial program 27.5

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

    3. Applied flip--27.6

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

    4. Simplified17.3

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

    5. Simplified17.3

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

    7. Applied *-un-lft-identity17.3

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

    8. Applied *-un-lft-identity17.3

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

    9. Applied *-un-lft-identity17.3

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

    10. Applied times-frac17.3

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

    11. Applied times-frac17.3

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

    12. Simplified17.3

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

    13. Simplified17.1

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

    14. Taylor expanded around 0 10.0

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

    if -2.851839700712213e-304 < b_2 < 4.71744724099961e+65

    1. Initial program 9.1

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

    3. Applied div-inv9.2

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

    if 4.71744724099961e+65 < b_2

    1. Initial program 38.1

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

    3. Applied flip--59.9

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

    4. Simplified60.1

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

    5. Simplified60.1

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

    6. Taylor expanded around 0 6.1

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

  4. Final simplification7.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -227369802444031.66:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -2.851839700712213 \cdot 10^{-304}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 4.71744724099961 \cdot 10^{+65}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))