Average Error: 34.0 → 10.6
Time: 4.1s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.731633690849517820308375807349583220341 \cdot 10^{-121}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.02738286211209785784187544728837722875 \cdot 10^{63}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.731633690849517820308375807349583220341 \cdot 10^{-121}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.02738286211209785784187544728837722875 \cdot 10^{63}:\\
\;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\

\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 r69691 = b_2;
        double r69692 = -r69691;
        double r69693 = r69691 * r69691;
        double r69694 = a;
        double r69695 = c;
        double r69696 = r69694 * r69695;
        double r69697 = r69693 - r69696;
        double r69698 = sqrt(r69697);
        double r69699 = r69692 - r69698;
        double r69700 = r69699 / r69694;
        return r69700;
}
↓
double f(double a, double b_2, double c) {
        double r69701 = b_2;
        double r69702 = -2.731633690849518e-121;
        bool r69703 = r69701 <= r69702;
        double r69704 = -0.5;
        double r69705 = c;
        double r69706 = r69705 / r69701;
        double r69707 = r69704 * r69706;
        double r69708 = 1.0273828621120979e+63;
        bool r69709 = r69701 <= r69708;
        double r69710 = 1.0;
        double r69711 = a;
        double r69712 = -r69701;
        double r69713 = r69701 * r69701;
        double r69714 = r69711 * r69705;
        double r69715 = r69713 - r69714;
        double r69716 = sqrt(r69715);
        double r69717 = r69712 - r69716;
        double r69718 = r69711 / r69717;
        double r69719 = r69710 / r69718;
        double r69720 = 0.5;
        double r69721 = r69720 * r69706;
        double r69722 = 2.0;
        double r69723 = r69701 / r69711;
        double r69724 = r69722 * r69723;
        double r69725 = r69721 - r69724;
        double r69726 = r69709 ? r69719 : r69725;
        double r69727 = r69703 ? r69707 : r69726;
        return r69727;
}

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 < -2.731633690849518e-121

    1. Initial program 51.0

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

    2. Taylor expanded around -inf 11.5

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

    if -2.731633690849518e-121 < b_2 < 1.0273828621120979e+63

    1. Initial program 12.1

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

    3. Applied clear-num12.2

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

    if 1.0273828621120979e+63 < b_2

    1. Initial program 39.8

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

    2. Taylor expanded around inf 5.4

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.731633690849517820308375807349583220341 \cdot 10^{-121}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.02738286211209785784187544728837722875 \cdot 10^{63}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))