Average Error: 34.3 → 10.2
Time: 6.5s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.4278904486834676 \cdot 10^{-42}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.8046284917653458 \cdot 10^{91}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.4278904486834676 \cdot 10^{-42}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.8046284917653458 \cdot 10^{91}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\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 r19448 = b_2;
        double r19449 = -r19448;
        double r19450 = r19448 * r19448;
        double r19451 = a;
        double r19452 = c;
        double r19453 = r19451 * r19452;
        double r19454 = r19450 - r19453;
        double r19455 = sqrt(r19454);
        double r19456 = r19449 - r19455;
        double r19457 = r19456 / r19451;
        return r19457;
}
↓
double f(double a, double b_2, double c) {
        double r19458 = b_2;
        double r19459 = -5.4278904486834676e-42;
        bool r19460 = r19458 <= r19459;
        double r19461 = -0.5;
        double r19462 = c;
        double r19463 = r19462 / r19458;
        double r19464 = r19461 * r19463;
        double r19465 = 2.8046284917653458e+91;
        bool r19466 = r19458 <= r19465;
        double r19467 = -r19458;
        double r19468 = r19458 * r19458;
        double r19469 = a;
        double r19470 = r19469 * r19462;
        double r19471 = r19468 - r19470;
        double r19472 = sqrt(r19471);
        double r19473 = r19467 - r19472;
        double r19474 = r19473 / r19469;
        double r19475 = 0.5;
        double r19476 = r19475 * r19463;
        double r19477 = 2.0;
        double r19478 = r19458 / r19469;
        double r19479 = r19477 * r19478;
        double r19480 = r19476 - r19479;
        double r19481 = r19466 ? r19474 : r19480;
        double r19482 = r19460 ? r19464 : r19481;
        return r19482;
}

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 < -5.4278904486834676e-42

    1. Initial program 54.7

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

    2. Taylor expanded around -inf 7.1

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

    if -5.4278904486834676e-42 < b_2 < 2.8046284917653458e+91

    1. Initial program 14.7

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

    if 2.8046284917653458e+91 < b_2

    1. Initial program 45.3

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

    2. Taylor expanded around inf 4.1

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.4278904486834676 \cdot 10^{-42}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.8046284917653458 \cdot 10^{91}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +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))