Average Error: 34.1 → 6.7
Time: 5.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.478383220944118 \cdot 10^{90}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 1.071982619004943 \cdot 10^{-308}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{1}{3 \cdot a}\\ \mathbf{elif}\;b \le 1.8676563684114658 \cdot 10^{102}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{c}{1}} \cdot \left(\left(-b\right) - \sqrt{b \cdot b + \left(0 - 3 \cdot \left(a \cdot c\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 -7.478383220944118 \cdot 10^{90}:\\
\;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\

\mathbf{elif}\;b \le 1.071982619004943 \cdot 10^{-308}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{1}{3 \cdot a}\\

\mathbf{elif}\;b \le 1.8676563684114658 \cdot 10^{102}:\\
\;\;\;\;\frac{1}{\frac{1}{\frac{c}{1}} \cdot \left(\left(-b\right) - \sqrt{b \cdot b + \left(0 - 3 \cdot \left(a \cdot c\right)\right)}\right)}\\

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

\end{array}
double f(double a, double b, double c) {
        double r123969 = b;
        double r123970 = -r123969;
        double r123971 = r123969 * r123969;
        double r123972 = 3.0;
        double r123973 = a;
        double r123974 = r123972 * r123973;
        double r123975 = c;
        double r123976 = r123974 * r123975;
        double r123977 = r123971 - r123976;
        double r123978 = sqrt(r123977);
        double r123979 = r123970 + r123978;
        double r123980 = r123979 / r123974;
        return r123980;
}
↓
double f(double a, double b, double c) {
        double r123981 = b;
        double r123982 = -7.478383220944118e+90;
        bool r123983 = r123981 <= r123982;
        double r123984 = 0.5;
        double r123985 = c;
        double r123986 = r123985 / r123981;
        double r123987 = r123984 * r123986;
        double r123988 = 0.6666666666666666;
        double r123989 = a;
        double r123990 = r123981 / r123989;
        double r123991 = r123988 * r123990;
        double r123992 = r123987 - r123991;
        double r123993 = 1.0719826190049434e-308;
        bool r123994 = r123981 <= r123993;
        double r123995 = -r123981;
        double r123996 = r123981 * r123981;
        double r123997 = 3.0;
        double r123998 = r123997 * r123989;
        double r123999 = r123998 * r123985;
        double r124000 = r123996 - r123999;
        double r124001 = sqrt(r124000);
        double r124002 = r123995 + r124001;
        double r124003 = 1.0;
        double r124004 = r124003 / r123998;
        double r124005 = r124002 * r124004;
        double r124006 = 1.8676563684114658e+102;
        bool r124007 = r123981 <= r124006;
        double r124008 = r123985 / r124003;
        double r124009 = r124003 / r124008;
        double r124010 = 0.0;
        double r124011 = r123989 * r123985;
        double r124012 = r123997 * r124011;
        double r124013 = r124010 - r124012;
        double r124014 = r123996 + r124013;
        double r124015 = sqrt(r124014);
        double r124016 = r123995 - r124015;
        double r124017 = r124009 * r124016;
        double r124018 = r124003 / r124017;
        double r124019 = -0.5;
        double r124020 = r124019 * r123986;
        double r124021 = r124007 ? r124018 : r124020;
        double r124022 = r123994 ? r124005 : r124021;
        double r124023 = r123983 ? r123992 : r124022;
        return r124023;
}

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 < -7.478383220944118e+90

    1. Initial program 44.9

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

    2. Taylor expanded around -inf 3.7

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

    if -7.478383220944118e+90 < b < 1.0719826190049434e-308

    1. Initial program 9.2

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

    3. Applied div-inv9.2

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

    if 1.0719826190049434e-308 < b < 1.8676563684114658e+102

    1. Initial program 33.5

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

    3. Applied flip-+33.5

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

    4. Simplified17.3

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

    6. Applied sub-neg17.3

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

    7. Simplified17.3

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

    9. Applied clear-num17.5

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

    10. Simplified16.5

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

    12. Applied clear-num16.5

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

    13. Simplified9.2

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

    if 1.8676563684114658e+102 < b

    1. Initial program 59.8

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

    2. Taylor expanded around inf 2.4

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

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.478383220944118 \cdot 10^{90}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 1.071982619004943 \cdot 10^{-308}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{1}{3 \cdot a}\\ \mathbf{elif}\;b \le 1.8676563684114658 \cdot 10^{102}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{c}{1}} \cdot \left(\left(-b\right) - \sqrt{b \cdot b + \left(0 - 3 \cdot \left(a \cdot c\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))