Average Error: 34.1 → 8.1
Time: 6.2s
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 r109119 = b;
        double r109120 = -r109119;
        double r109121 = r109119 * r109119;
        double r109122 = 3.0;
        double r109123 = a;
        double r109124 = r109122 * r109123;
        double r109125 = c;
        double r109126 = r109124 * r109125;
        double r109127 = r109121 - r109126;
        double r109128 = sqrt(r109127);
        double r109129 = r109120 + r109128;
        double r109130 = r109129 / r109124;
        return r109130;
}
↓
double f(double a, double b, double c) {
        double r109131 = b;
        double r109132 = -1.2574766781276779e+107;
        bool r109133 = r109131 <= r109132;
        double r109134 = 0.5;
        double r109135 = c;
        double r109136 = r109135 / r109131;
        double r109137 = r109134 * r109136;
        double r109138 = 0.6666666666666666;
        double r109139 = a;
        double r109140 = r109131 / r109139;
        double r109141 = r109138 * r109140;
        double r109142 = r109137 - r109141;
        double r109143 = 2.4967447941335268e-135;
        bool r109144 = r109131 <= r109143;
        double r109145 = -r109131;
        double r109146 = r109131 * r109131;
        double r109147 = 3.0;
        double r109148 = r109147 * r109139;
        double r109149 = r109148 * r109135;
        double r109150 = r109146 - r109149;
        double r109151 = sqrt(r109150);
        double r109152 = r109145 + r109151;
        double r109153 = r109152 / r109147;
        double r109154 = r109153 / r109139;
        double r109155 = 5.354608489416471e+62;
        bool r109156 = r109131 <= r109155;
        double r109157 = cbrt(r109139);
        double r109158 = r109139 / r109157;
        double r109159 = r109158 / r109157;
        double r109160 = 1.0;
        double r109161 = r109145 - r109151;
        double r109162 = r109161 / r109135;
        double r109163 = r109160 / r109162;
        double r109164 = r109163 / r109157;
        double r109165 = r109159 * r109164;
        double r109166 = -0.5;
        double r109167 = r109166 * r109136;
        double r109168 = r109156 ? r109165 : r109167;
        double r109169 = r109144 ? r109154 : r109168;
        double r109170 = r109133 ? r109142 : r109169;
        return r109170;
}

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