Average Error: 33.9 → 7.1
Time: 8.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.594947000714855189120603839967237527365 \cdot 10^{98}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{6004799503160661}{9007199254740992} \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le \frac{-5177063461970013}{2.415335951885786465434594931832492090187 \cdot 10^{170}}:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}\\ \mathbf{elif}\;b \le 2.421728556744193807684070877075700879531 \cdot 10^{61}:\\ \;\;\;\;\frac{\frac{c}{1}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 -8.594947000714855189120603839967237527365 \cdot 10^{98}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{6004799503160661}{9007199254740992} \cdot \frac{b}{a}\\

\mathbf{elif}\;b \le \frac{-5177063461970013}{2.415335951885786465434594931832492090187 \cdot 10^{170}}:\\
\;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}\\

\mathbf{elif}\;b \le 2.421728556744193807684070877075700879531 \cdot 10^{61}:\\
\;\;\;\;\frac{\frac{c}{1}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r115612 = b;
        double r115613 = -r115612;
        double r115614 = r115612 * r115612;
        double r115615 = 3.0;
        double r115616 = a;
        double r115617 = r115615 * r115616;
        double r115618 = c;
        double r115619 = r115617 * r115618;
        double r115620 = r115614 - r115619;
        double r115621 = sqrt(r115620);
        double r115622 = r115613 + r115621;
        double r115623 = r115622 / r115617;
        return r115623;
}
↓
double f(double a, double b, double c) {
        double r115624 = b;
        double r115625 = -8.594947000714855e+98;
        bool r115626 = r115624 <= r115625;
        double r115627 = 1.0;
        double r115628 = 2.0;
        double r115629 = r115627 / r115628;
        double r115630 = c;
        double r115631 = r115630 / r115624;
        double r115632 = r115629 * r115631;
        double r115633 = 6004799503160661.0;
        double r115634 = 9007199254740992.0;
        double r115635 = r115633 / r115634;
        double r115636 = a;
        double r115637 = r115624 / r115636;
        double r115638 = r115635 * r115637;
        double r115639 = r115632 - r115638;
        double r115640 = -5177063461970013.0;
        double r115641 = 2.4153359518857865e+170;
        double r115642 = r115640 / r115641;
        bool r115643 = r115624 <= r115642;
        double r115644 = -r115624;
        double r115645 = r115624 * r115624;
        double r115646 = 3.0;
        double r115647 = r115646 * r115636;
        double r115648 = r115647 * r115630;
        double r115649 = r115645 - r115648;
        double r115650 = sqrt(r115649);
        double r115651 = r115644 + r115650;
        double r115652 = r115651 / r115646;
        double r115653 = r115652 / r115636;
        double r115654 = 2.4217285567441938e+61;
        bool r115655 = r115624 <= r115654;
        double r115656 = r115630 / r115627;
        double r115657 = r115644 - r115650;
        double r115658 = r115656 / r115657;
        double r115659 = -1.0;
        double r115660 = r115659 / r115628;
        double r115661 = r115660 * r115631;
        double r115662 = r115655 ? r115658 : r115661;
        double r115663 = r115643 ? r115653 : r115662;
        double r115664 = r115626 ? r115639 : r115663;
        return r115664;
}

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 < -8.594947000714855e+98

    1. Initial program 46.3

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

    2. Taylor expanded around -inf 4.0

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b} - 0.6666666666666666296592325124947819858789 \cdot \frac{b}{a}}\]

    3. Simplified4.0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b} - \frac{6004799503160661}{9007199254740992} \cdot \frac{b}{a}}\]

    if -8.594947000714855e+98 < b < -2.1434134071196154e-155

    1. Initial program 6.2

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

    3. Applied associate-/r*6.2

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

    if -2.1434134071196154e-155 < b < 2.4217285567441938e+61

    1. Initial program 26.2

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

    3. Applied flip-+26.6

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

    4. Simplified16.6

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

    6. Applied *-un-lft-identity16.6

      \[\leadsto \frac{\frac{0 + 3 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}}{3 \cdot a}\]

    7. Applied *-un-lft-identity16.6

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(0 + 3 \cdot \left(a \cdot c\right)\right)}}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a}\]

    8. Applied times-frac16.6

      \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{0 + 3 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

    9. Applied associate-/l*16.7

      \[\leadsto \color{blue}{\frac{\frac{1}{1}}{\frac{3 \cdot a}{\frac{0 + 3 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}}\]

    10. Simplified16.5

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

    11. Taylor expanded around 0 11.4

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

    13. Applied associate-/r*11.3

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

    14. Simplified11.2

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

    if 2.4217285567441938e+61 < b

    1. Initial program 57.4

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

    2. Taylor expanded around inf 4.0

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

    3. Simplified4.0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification7.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.594947000714855189120603839967237527365 \cdot 10^{98}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{6004799503160661}{9007199254740992} \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le \frac{-5177063461970013}{2.415335951885786465434594931832492090187 \cdot 10^{170}}:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}\\ \mathbf{elif}\;b \le 2.421728556744193807684070877075700879531 \cdot 10^{61}:\\ \;\;\;\;\frac{\frac{c}{1}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))