Average Error: 33.7 → 9.0
Time: 19.9s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -6239046376.848015:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -2.915349047648131 \cdot 10^{-265}:\\ \;\;\;\;\frac{\frac{a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 4.71744724099961 \cdot 10^{+65}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -6239046376.848015:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -2.915349047648131 \cdot 10^{-265}:\\
\;\;\;\;\frac{\frac{a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\\

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

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

\end{array}
double f(double a, double b_2, double c) {
        double r1932091 = b_2;
        double r1932092 = -r1932091;
        double r1932093 = r1932091 * r1932091;
        double r1932094 = a;
        double r1932095 = c;
        double r1932096 = r1932094 * r1932095;
        double r1932097 = r1932093 - r1932096;
        double r1932098 = sqrt(r1932097);
        double r1932099 = r1932092 - r1932098;
        double r1932100 = r1932099 / r1932094;
        return r1932100;
}
↓
double f(double a, double b_2, double c) {
        double r1932101 = b_2;
        double r1932102 = -6239046376.848015;
        bool r1932103 = r1932101 <= r1932102;
        double r1932104 = -0.5;
        double r1932105 = c;
        double r1932106 = r1932105 / r1932101;
        double r1932107 = r1932104 * r1932106;
        double r1932108 = -2.915349047648131e-265;
        bool r1932109 = r1932101 <= r1932108;
        double r1932110 = a;
        double r1932111 = r1932110 * r1932105;
        double r1932112 = r1932101 * r1932101;
        double r1932113 = r1932112 - r1932112;
        double r1932114 = r1932111 + r1932113;
        double r1932115 = r1932112 - r1932111;
        double r1932116 = sqrt(r1932115);
        double r1932117 = r1932116 - r1932101;
        double r1932118 = r1932114 / r1932117;
        double r1932119 = r1932118 / r1932110;
        double r1932120 = 4.71744724099961e+65;
        bool r1932121 = r1932101 <= r1932120;
        double r1932122 = 1.0;
        double r1932123 = r1932122 / r1932110;
        double r1932124 = -r1932101;
        double r1932125 = r1932124 - r1932116;
        double r1932126 = r1932123 * r1932125;
        double r1932127 = r1932101 / r1932110;
        double r1932128 = -2.0;
        double r1932129 = r1932127 * r1932128;
        double r1932130 = r1932121 ? r1932126 : r1932129;
        double r1932131 = r1932109 ? r1932119 : r1932130;
        double r1932132 = r1932103 ? r1932107 : r1932131;
        return r1932132;
}

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 < -6239046376.848015

    1. Initial program 55.7

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

    2. Taylor expanded around -inf 5.1

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

    if -6239046376.848015 < b_2 < -2.915349047648131e-265

    1. Initial program 28.7

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

    3. Applied flip--28.8

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

    4. Simplified17.5

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

    5. Simplified17.5

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

    if -2.915349047648131e-265 < b_2 < 4.71744724099961e+65

    1. Initial program 9.5

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

    3. Applied div-inv9.7

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

    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 clear-num38.2

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

    4. Taylor expanded around 0 6.0

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

  4. Final simplification9.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -6239046376.848015:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -2.915349047648131 \cdot 10^{-265}:\\ \;\;\;\;\frac{\frac{a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 4.71744724099961 \cdot 10^{+65}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array}\]

Reproduce

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