Average Error: 33.4 → 15.4
Time: 16.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.277637730923319 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{3}{2} \cdot \frac{a \cdot c}{b} - 2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \le 3.32629031803127 \cdot 10^{-71}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}{a \cdot 3}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.277637730923319 \cdot 10^{+112}:\\
\;\;\;\;\frac{\frac{3}{2} \cdot \frac{a \cdot c}{b} - 2 \cdot b}{a \cdot 3}\\

\mathbf{elif}\;b \le 3.32629031803127 \cdot 10^{-71}:\\
\;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}} - b}{a \cdot 3}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}{a \cdot 3}\\

\end{array}
double f(double a, double b, double c) {
        double r1744431 = b;
        double r1744432 = -r1744431;
        double r1744433 = r1744431 * r1744431;
        double r1744434 = 3.0;
        double r1744435 = a;
        double r1744436 = r1744434 * r1744435;
        double r1744437 = c;
        double r1744438 = r1744436 * r1744437;
        double r1744439 = r1744433 - r1744438;
        double r1744440 = sqrt(r1744439);
        double r1744441 = r1744432 + r1744440;
        double r1744442 = r1744441 / r1744436;
        return r1744442;
}
↓
double f(double a, double b, double c) {
        double r1744443 = b;
        double r1744444 = -1.277637730923319e+112;
        bool r1744445 = r1744443 <= r1744444;
        double r1744446 = 1.5;
        double r1744447 = a;
        double r1744448 = c;
        double r1744449 = r1744447 * r1744448;
        double r1744450 = r1744449 / r1744443;
        double r1744451 = r1744446 * r1744450;
        double r1744452 = 2.0;
        double r1744453 = r1744452 * r1744443;
        double r1744454 = r1744451 - r1744453;
        double r1744455 = 3.0;
        double r1744456 = r1744447 * r1744455;
        double r1744457 = r1744454 / r1744456;
        double r1744458 = 3.32629031803127e-71;
        bool r1744459 = r1744443 <= r1744458;
        double r1744460 = r1744443 * r1744443;
        double r1744461 = r1744455 * r1744449;
        double r1744462 = r1744460 - r1744461;
        double r1744463 = sqrt(r1744462);
        double r1744464 = sqrt(r1744463);
        double r1744465 = r1744464 * r1744464;
        double r1744466 = r1744465 - r1744443;
        double r1744467 = r1744466 / r1744456;
        double r1744468 = -1.5;
        double r1744469 = r1744468 * r1744450;
        double r1744470 = r1744469 / r1744456;
        double r1744471 = r1744459 ? r1744467 : r1744470;
        double r1744472 = r1744445 ? r1744457 : r1744471;
        return r1744472;
}

Error

Bits error versus a

Bits error versus b

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 < -1.277637730923319e+112

    1. Initial program 47.4

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

    2. Simplified47.4

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}}\]

    3. Taylor expanded around 0 47.4

      \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}} - b}{3 \cdot a}\]

    4. Taylor expanded around -inf 9.9

      \[\leadsto \frac{\color{blue}{\frac{3}{2} \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{3 \cdot a}\]

    if -1.277637730923319e+112 < b < 3.32629031803127e-71

    1. Initial program 13.0

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

    2. Simplified13.0

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}}\]

    3. Taylor expanded around 0 13.0

      \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}} - b}{3 \cdot a}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt13.2

      \[\leadsto \frac{\color{blue}{\sqrt{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}} - b}{3 \cdot a}\]

    if 3.32629031803127e-71 < b

    1. Initial program 52.6

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

    2. Simplified52.6

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}}\]

    3. Taylor expanded around inf 19.9

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

  4. Final simplification15.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.277637730923319 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{3}{2} \cdot \frac{a \cdot c}{b} - 2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \le 3.32629031803127 \cdot 10^{-71}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}{a \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))