Average Error: 34.0 → 10.6
Time: 4.5s
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 r12384 = b_2;
        double r12385 = -r12384;
        double r12386 = r12384 * r12384;
        double r12387 = a;
        double r12388 = c;
        double r12389 = r12387 * r12388;
        double r12390 = r12386 - r12389;
        double r12391 = sqrt(r12390);
        double r12392 = r12385 - r12391;
        double r12393 = r12392 / r12387;
        return r12393;
}
↓
double f(double a, double b_2, double c) {
        double r12394 = b_2;
        double r12395 = -2.731633690849518e-121;
        bool r12396 = r12394 <= r12395;
        double r12397 = -0.5;
        double r12398 = c;
        double r12399 = r12398 / r12394;
        double r12400 = r12397 * r12399;
        double r12401 = 1.0273828621120979e+63;
        bool r12402 = r12394 <= r12401;
        double r12403 = 1.0;
        double r12404 = a;
        double r12405 = -r12394;
        double r12406 = r12394 * r12394;
        double r12407 = r12404 * r12398;
        double r12408 = r12406 - r12407;
        double r12409 = sqrt(r12408);
        double r12410 = r12405 - r12409;
        double r12411 = r12404 / r12410;
        double r12412 = r12403 / r12411;
        double r12413 = 0.5;
        double r12414 = r12413 * r12399;
        double r12415 = 2.0;
        double r12416 = r12394 / r12404;
        double r12417 = r12415 * r12416;
        double r12418 = r12414 - r12417;
        double r12419 = r12402 ? r12412 : r12418;
        double r12420 = r12396 ? r12400 : r12419;
        return r12420;
}

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 "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))