Average Error: 34.2 → 10.4
Time: 4.8s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.4270058556435274 \cdot 10^{-117}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.49922826628406174 \cdot 10^{84}:\\ \;\;\;\;1 \cdot \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -4.4270058556435274 \cdot 10^{-117}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.49922826628406174 \cdot 10^{84}:\\
\;\;\;\;1 \cdot \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 r76534 = b_2;
        double r76535 = -r76534;
        double r76536 = r76534 * r76534;
        double r76537 = a;
        double r76538 = c;
        double r76539 = r76537 * r76538;
        double r76540 = r76536 - r76539;
        double r76541 = sqrt(r76540);
        double r76542 = r76535 - r76541;
        double r76543 = r76542 / r76537;
        return r76543;
}
↓
double f(double a, double b_2, double c) {
        double r76544 = b_2;
        double r76545 = -4.4270058556435274e-117;
        bool r76546 = r76544 <= r76545;
        double r76547 = -0.5;
        double r76548 = c;
        double r76549 = r76548 / r76544;
        double r76550 = r76547 * r76549;
        double r76551 = 2.4992282662840617e+84;
        bool r76552 = r76544 <= r76551;
        double r76553 = 1.0;
        double r76554 = -r76544;
        double r76555 = r76544 * r76544;
        double r76556 = a;
        double r76557 = r76556 * r76548;
        double r76558 = r76555 - r76557;
        double r76559 = sqrt(r76558);
        double r76560 = r76554 - r76559;
        double r76561 = r76560 / r76556;
        double r76562 = r76553 * r76561;
        double r76563 = 0.5;
        double r76564 = r76563 * r76549;
        double r76565 = 2.0;
        double r76566 = r76544 / r76556;
        double r76567 = r76565 * r76566;
        double r76568 = r76564 - r76567;
        double r76569 = r76552 ? r76562 : r76568;
        double r76570 = r76546 ? r76550 : r76569;
        return r76570;
}

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 < -4.4270058556435274e-117

    1. Initial program 51.5

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

    2. Taylor expanded around -inf 11.0

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

    if -4.4270058556435274e-117 < b_2 < 2.4992282662840617e+84

    1. Initial program 12.4

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

    3. Applied div-inv12.5

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

    5. Applied *-un-lft-identity12.5

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

    6. Applied associate-*l*12.5

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

    7. Simplified12.4

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

    if 2.4992282662840617e+84 < b_2

    1. Initial program 43.1

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

    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 simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.4270058556435274 \cdot 10^{-117}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.49922826628406174 \cdot 10^{84}:\\ \;\;\;\;1 \cdot \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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))