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 r66676 = b_2;
        double r66677 = -r66676;
        double r66678 = r66676 * r66676;
        double r66679 = a;
        double r66680 = c;
        double r66681 = r66679 * r66680;
        double r66682 = r66678 - r66681;
        double r66683 = sqrt(r66682);
        double r66684 = r66677 - r66683;
        double r66685 = r66684 / r66679;
        return r66685;
}
↓
double f(double a, double b_2, double c) {
        double r66686 = b_2;
        double r66687 = -2.731633690849518e-121;
        bool r66688 = r66686 <= r66687;
        double r66689 = -0.5;
        double r66690 = c;
        double r66691 = r66690 / r66686;
        double r66692 = r66689 * r66691;
        double r66693 = 1.0273828621120979e+63;
        bool r66694 = r66686 <= r66693;
        double r66695 = 1.0;
        double r66696 = a;
        double r66697 = -r66686;
        double r66698 = r66686 * r66686;
        double r66699 = r66696 * r66690;
        double r66700 = r66698 - r66699;
        double r66701 = sqrt(r66700);
        double r66702 = r66697 - r66701;
        double r66703 = r66696 / r66702;
        double r66704 = r66695 / r66703;
        double r66705 = 0.5;
        double r66706 = r66705 * r66691;
        double r66707 = 2.0;
        double r66708 = r66686 / r66696;
        double r66709 = r66707 * r66708;
        double r66710 = r66706 - r66709;
        double r66711 = r66694 ? r66704 : r66710;
        double r66712 = r66688 ? r66692 : r66711;
        return r66712;
}

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))