Average Error: 34.1 → 8.1
Time: 6.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.257476678127677856918278287038350045718 \cdot 10^{107}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666296592325124947819858789 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 2.496744794133526836762101371765290843051 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}\\ \mathbf{elif}\;b \le 5.354608489416471204042085887246325611474 \cdot 10^{62}:\\ \;\;\;\;\frac{\frac{a}{\sqrt[3]{a}}}{\sqrt[3]{a}} \cdot \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{c}}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.257476678127677856918278287038350045718 \cdot 10^{107}:\\
\;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666296592325124947819858789 \cdot \frac{b}{a}\\

\mathbf{elif}\;b \le 2.496744794133526836762101371765290843051 \cdot 10^{-135}:\\
\;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}\\

\mathbf{elif}\;b \le 5.354608489416471204042085887246325611474 \cdot 10^{62}:\\
\;\;\;\;\frac{\frac{a}{\sqrt[3]{a}}}{\sqrt[3]{a}} \cdot \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{c}}}{\sqrt[3]{a}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r123042 = b;
        double r123043 = -r123042;
        double r123044 = r123042 * r123042;
        double r123045 = 3.0;
        double r123046 = a;
        double r123047 = r123045 * r123046;
        double r123048 = c;
        double r123049 = r123047 * r123048;
        double r123050 = r123044 - r123049;
        double r123051 = sqrt(r123050);
        double r123052 = r123043 + r123051;
        double r123053 = r123052 / r123047;
        return r123053;
}
↓
double f(double a, double b, double c) {
        double r123054 = b;
        double r123055 = -1.2574766781276779e+107;
        bool r123056 = r123054 <= r123055;
        double r123057 = 0.5;
        double r123058 = c;
        double r123059 = r123058 / r123054;
        double r123060 = r123057 * r123059;
        double r123061 = 0.6666666666666666;
        double r123062 = a;
        double r123063 = r123054 / r123062;
        double r123064 = r123061 * r123063;
        double r123065 = r123060 - r123064;
        double r123066 = 2.4967447941335268e-135;
        bool r123067 = r123054 <= r123066;
        double r123068 = -r123054;
        double r123069 = r123054 * r123054;
        double r123070 = 3.0;
        double r123071 = r123070 * r123062;
        double r123072 = r123071 * r123058;
        double r123073 = r123069 - r123072;
        double r123074 = sqrt(r123073);
        double r123075 = r123068 + r123074;
        double r123076 = r123075 / r123070;
        double r123077 = r123076 / r123062;
        double r123078 = 5.354608489416471e+62;
        bool r123079 = r123054 <= r123078;
        double r123080 = cbrt(r123062);
        double r123081 = r123062 / r123080;
        double r123082 = r123081 / r123080;
        double r123083 = 1.0;
        double r123084 = r123068 - r123074;
        double r123085 = r123084 / r123058;
        double r123086 = r123083 / r123085;
        double r123087 = r123086 / r123080;
        double r123088 = r123082 * r123087;
        double r123089 = -0.5;
        double r123090 = r123089 * r123059;
        double r123091 = r123079 ? r123088 : r123090;
        double r123092 = r123067 ? r123077 : r123091;
        double r123093 = r123056 ? r123065 : r123092;
        return r123093;
}

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 < -1.2574766781276779e+107

    1. Initial program 48.2

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

    2. Taylor expanded around -inf 3.7

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

    if -1.2574766781276779e+107 < b < 2.4967447941335268e-135

    1. Initial program 11.7

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

    3. Applied associate-/r*11.7

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

    if 2.4967447941335268e-135 < b < 5.354608489416471e+62

    1. Initial program 39.0

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

    3. Applied flip-+39.0

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

    4. Simplified16.1

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

    6. Applied associate-/r*16.1

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

    7. Simplified16.1

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

    9. Applied clear-num16.4

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

    10. Simplified16.3

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

    12. Applied add-cube-cbrt17.0

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

    13. Applied *-un-lft-identity17.0

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

    14. Applied times-frac14.3

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

    15. Applied add-sqr-sqrt14.3

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

    16. Applied times-frac13.7

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

    17. Applied times-frac10.2

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

    18. Simplified10.2

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

    19. Simplified10.2

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

    if 5.354608489416471e+62 < b

    1. Initial program 57.7

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

    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 simplification8.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.257476678127677856918278287038350045718 \cdot 10^{107}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666296592325124947819858789 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 2.496744794133526836762101371765290843051 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}\\ \mathbf{elif}\;b \le 5.354608489416471204042085887246325611474 \cdot 10^{62}:\\ \;\;\;\;\frac{\frac{a}{\sqrt[3]{a}}}{\sqrt[3]{a}} \cdot \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{c}}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))