Average Error: 32.9 → 14.8
Time: 18.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.3376190644449892 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(\frac{3}{2} \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \le 6.235673785124529 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]
\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 -1.3376190644449892 \cdot 10^{+154}:\\
\;\;\;\;\frac{\left(\frac{3}{2} \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\

\mathbf{elif}\;b \le 6.235673785124529 \cdot 10^{-73}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1628643 = b;
        double r1628644 = -r1628643;
        double r1628645 = r1628643 * r1628643;
        double r1628646 = 3.0;
        double r1628647 = a;
        double r1628648 = r1628646 * r1628647;
        double r1628649 = c;
        double r1628650 = r1628648 * r1628649;
        double r1628651 = r1628645 - r1628650;
        double r1628652 = sqrt(r1628651);
        double r1628653 = r1628644 + r1628652;
        double r1628654 = r1628653 / r1628648;
        return r1628654;
}
↓
double f(double a, double b, double c) {
        double r1628655 = b;
        double r1628656 = -1.3376190644449892e+154;
        bool r1628657 = r1628655 <= r1628656;
        double r1628658 = 1.5;
        double r1628659 = a;
        double r1628660 = c;
        double r1628661 = r1628659 * r1628660;
        double r1628662 = r1628661 / r1628655;
        double r1628663 = r1628658 * r1628662;
        double r1628664 = r1628663 - r1628655;
        double r1628665 = r1628664 - r1628655;
        double r1628666 = 3.0;
        double r1628667 = r1628666 * r1628659;
        double r1628668 = r1628665 / r1628667;
        double r1628669 = 6.235673785124529e-73;
        bool r1628670 = r1628655 <= r1628669;
        double r1628671 = r1628655 * r1628655;
        double r1628672 = r1628666 * r1628661;
        double r1628673 = r1628671 - r1628672;
        double r1628674 = sqrt(r1628673);
        double r1628675 = r1628674 - r1628655;
        double r1628676 = r1628675 / r1628667;
        double r1628677 = -1.5;
        double r1628678 = r1628677 * r1628662;
        double r1628679 = r1628678 / r1628667;
        double r1628680 = r1628670 ? r1628676 : r1628679;
        double r1628681 = r1628657 ? r1628668 : r1628680;
        return r1628681;
}

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 < -1.3376190644449892e+154

    1. Initial program 60.9

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

    2. Simplified60.9

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

    4. Applied associate-*l*60.9

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

    5. Taylor expanded around -inf 10.6

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

    if -1.3376190644449892e+154 < b < 6.235673785124529e-73

    1. Initial program 12.0

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

    2. Simplified12.0

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

    4. Applied associate-*l*12.1

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

    if 6.235673785124529e-73 < b

    1. Initial program 52.3

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

    2. Simplified52.3

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

    4. Applied add-sqr-sqrt52.3

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

    5. Applied sqrt-prod54.4

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

    6. Taylor expanded around inf 19.7

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

  4. Final simplification14.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3376190644449892 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(\frac{3}{2} \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \le 6.235673785124529 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))