Average Error: 34.5 → 9.7
Time: 4.7s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.9358923729233266 \cdot 10^{149}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 9.39036747108992214 \cdot 10^{-69}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.9358923729233266 \cdot 10^{149}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 9.39036747108992214 \cdot 10^{-69}:\\
\;\;\;\;\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}\\

\end{array}
double f(double a, double b_2, double c) {
        double r14590 = b_2;
        double r14591 = -r14590;
        double r14592 = r14590 * r14590;
        double r14593 = a;
        double r14594 = c;
        double r14595 = r14593 * r14594;
        double r14596 = r14592 - r14595;
        double r14597 = sqrt(r14596);
        double r14598 = r14591 + r14597;
        double r14599 = r14598 / r14593;
        return r14599;
}
↓
double f(double a, double b_2, double c) {
        double r14600 = b_2;
        double r14601 = -2.9358923729233266e+149;
        bool r14602 = r14600 <= r14601;
        double r14603 = 0.5;
        double r14604 = c;
        double r14605 = r14604 / r14600;
        double r14606 = r14603 * r14605;
        double r14607 = 2.0;
        double r14608 = a;
        double r14609 = r14600 / r14608;
        double r14610 = r14607 * r14609;
        double r14611 = r14606 - r14610;
        double r14612 = 9.390367471089922e-69;
        bool r14613 = r14600 <= r14612;
        double r14614 = -r14600;
        double r14615 = r14600 * r14600;
        double r14616 = r14608 * r14604;
        double r14617 = r14615 - r14616;
        double r14618 = sqrt(r14617);
        double r14619 = r14614 + r14618;
        double r14620 = r14619 / r14608;
        double r14621 = -0.5;
        double r14622 = r14621 * r14605;
        double r14623 = r14613 ? r14620 : r14622;
        double r14624 = r14602 ? r14611 : r14623;
        return r14624;
}

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 < -2.9358923729233266e+149

    1. Initial program 62.1

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

    2. Taylor expanded around -inf 1.7

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

    if -2.9358923729233266e+149 < b_2 < 9.390367471089922e-69

    1. Initial program 12.4

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

    if 9.390367471089922e-69 < b_2

    1. Initial program 53.5

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

    2. Taylor expanded around inf 8.7

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

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.9358923729233266 \cdot 10^{149}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 9.39036747108992214 \cdot 10^{-69}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))