Average Error: 34.3 → 8.8
Time: 4.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.55528137777049654 \cdot 10^{140}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 2.5319914608763346 \cdot 10^{-199}:\\ \;\;\;\;\frac{1}{\frac{3}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{a}}}\\ \mathbf{elif}\;b \le 6.8279232074078283 \cdot 10^{44}:\\ \;\;\;\;\frac{\frac{\frac{0 + 3 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 -8.55528137777049654 \cdot 10^{140}:\\
\;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\

\mathbf{elif}\;b \le 2.5319914608763346 \cdot 10^{-199}:\\
\;\;\;\;\frac{1}{\frac{3}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{a}}}\\

\mathbf{elif}\;b \le 6.8279232074078283 \cdot 10^{44}:\\
\;\;\;\;\frac{\frac{\frac{0 + 3 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3}}{a}\\

\mathbf{else}:\\
\;\;\;\;-0.5 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r141674 = b;
        double r141675 = -r141674;
        double r141676 = r141674 * r141674;
        double r141677 = 3.0;
        double r141678 = a;
        double r141679 = r141677 * r141678;
        double r141680 = c;
        double r141681 = r141679 * r141680;
        double r141682 = r141676 - r141681;
        double r141683 = sqrt(r141682);
        double r141684 = r141675 + r141683;
        double r141685 = r141684 / r141679;
        return r141685;
}
↓
double f(double a, double b, double c) {
        double r141686 = b;
        double r141687 = -8.555281377770497e+140;
        bool r141688 = r141686 <= r141687;
        double r141689 = 0.5;
        double r141690 = c;
        double r141691 = r141690 / r141686;
        double r141692 = r141689 * r141691;
        double r141693 = 0.6666666666666666;
        double r141694 = a;
        double r141695 = r141686 / r141694;
        double r141696 = r141693 * r141695;
        double r141697 = r141692 - r141696;
        double r141698 = 2.5319914608763346e-199;
        bool r141699 = r141686 <= r141698;
        double r141700 = 1.0;
        double r141701 = 3.0;
        double r141702 = -r141686;
        double r141703 = r141686 * r141686;
        double r141704 = r141701 * r141694;
        double r141705 = r141704 * r141690;
        double r141706 = r141703 - r141705;
        double r141707 = sqrt(r141706);
        double r141708 = r141702 + r141707;
        double r141709 = r141708 / r141694;
        double r141710 = r141701 / r141709;
        double r141711 = r141700 / r141710;
        double r141712 = 6.827923207407828e+44;
        bool r141713 = r141686 <= r141712;
        double r141714 = 0.0;
        double r141715 = r141694 * r141690;
        double r141716 = r141701 * r141715;
        double r141717 = r141714 + r141716;
        double r141718 = r141702 - r141707;
        double r141719 = r141717 / r141718;
        double r141720 = r141719 / r141701;
        double r141721 = r141720 / r141694;
        double r141722 = -0.5;
        double r141723 = r141722 * r141691;
        double r141724 = r141713 ? r141721 : r141723;
        double r141725 = r141699 ? r141711 : r141724;
        double r141726 = r141688 ? r141697 : r141725;
        return r141726;
}

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 4 regimes

  2. if b < -8.555281377770497e+140

    1. Initial program 58.5

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

    2. Taylor expanded around -inf 2.8

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}}\]

    if -8.555281377770497e+140 < b < 2.5319914608763346e-199

    1. Initial program 10.6

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

    3. Applied clear-num10.7

      \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}\]
    4. Using strategy rm

    5. Applied associate-/l*10.7

      \[\leadsto \frac{1}{\color{blue}{\frac{3}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{a}}}}\]

    if 2.5319914608763346e-199 < b < 6.827923207407828e+44

    1. Initial program 31.7

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

    3. Applied associate-/r*31.8

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

    5. Applied flip-+31.8

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

    6. Simplified16.5

      \[\leadsto \frac{\frac{\frac{\color{blue}{0 + 3 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3}}{a}\]

    if 6.827923207407828e+44 < b

    1. Initial program 56.8

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

    2. Taylor expanded around inf 3.8

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.55528137777049654 \cdot 10^{140}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 2.5319914608763346 \cdot 10^{-199}:\\ \;\;\;\;\frac{1}{\frac{3}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{a}}}\\ \mathbf{elif}\;b \le 6.8279232074078283 \cdot 10^{44}:\\ \;\;\;\;\frac{\frac{\frac{0 + 3 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))