Average Error: 33.7 → 15.6
Time: 7.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.2724541866372811 \cdot 10^{165}:\\ \;\;\;\;\frac{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \le 3.50340075496625935 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{{b}^{2} - \left(\sqrt[3]{3 \cdot \left(a \cdot c\right)} \cdot \left(\sqrt[3]{3} \cdot \sqrt[3]{a \cdot c}\right)\right) \cdot \sqrt[3]{3 \cdot \left(a \cdot c\right)}} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]
\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 -2.2724541866372811 \cdot 10^{165}:\\
\;\;\;\;\frac{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\

\mathbf{elif}\;b \le 3.50340075496625935 \cdot 10^{-57}:\\
\;\;\;\;\frac{\sqrt{{b}^{2} - \left(\sqrt[3]{3 \cdot \left(a \cdot c\right)} \cdot \left(\sqrt[3]{3} \cdot \sqrt[3]{a \cdot c}\right)\right) \cdot \sqrt[3]{3 \cdot \left(a \cdot c\right)}} - b}{3 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\

\end{array}
double f(double a, double b, double c) {
        double r173084 = b;
        double r173085 = -r173084;
        double r173086 = r173084 * r173084;
        double r173087 = 3.0;
        double r173088 = a;
        double r173089 = r173087 * r173088;
        double r173090 = c;
        double r173091 = r173089 * r173090;
        double r173092 = r173086 - r173091;
        double r173093 = sqrt(r173092);
        double r173094 = r173085 + r173093;
        double r173095 = r173094 / r173089;
        return r173095;
}
↓
double f(double a, double b, double c) {
        double r173096 = b;
        double r173097 = -2.272454186637281e+165;
        bool r173098 = r173096 <= r173097;
        double r173099 = 1.5;
        double r173100 = a;
        double r173101 = c;
        double r173102 = r173100 * r173101;
        double r173103 = r173102 / r173096;
        double r173104 = r173099 * r173103;
        double r173105 = r173104 - r173096;
        double r173106 = r173105 - r173096;
        double r173107 = 3.0;
        double r173108 = r173107 * r173100;
        double r173109 = r173106 / r173108;
        double r173110 = 3.5034007549662594e-57;
        bool r173111 = r173096 <= r173110;
        double r173112 = 2.0;
        double r173113 = pow(r173096, r173112);
        double r173114 = r173107 * r173102;
        double r173115 = cbrt(r173114);
        double r173116 = cbrt(r173107);
        double r173117 = cbrt(r173102);
        double r173118 = r173116 * r173117;
        double r173119 = r173115 * r173118;
        double r173120 = r173119 * r173115;
        double r173121 = r173113 - r173120;
        double r173122 = sqrt(r173121);
        double r173123 = r173122 - r173096;
        double r173124 = r173123 / r173108;
        double r173125 = -1.5;
        double r173126 = r173125 * r173103;
        double r173127 = r173126 / r173108;
        double r173128 = r173111 ? r173124 : r173127;
        double r173129 = r173098 ? r173109 : r173128;
        return r173129;
}

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 3 regimes

  2. if b < -2.272454186637281e+165

    1. Initial program 64.0

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

    2. Simplified64.0

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

    3. Taylor expanded around 0 64.0

      \[\leadsto \frac{\sqrt{\color{blue}{{b}^{2} - 3 \cdot \left(a \cdot c\right)}} - b}{3 \cdot a}\]

    4. Taylor expanded around -inf 10.5

      \[\leadsto \frac{\color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right)} - b}{3 \cdot a}\]

    if -2.272454186637281e+165 < b < 3.5034007549662594e-57

    1. Initial program 13.3

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

    2. Simplified13.3

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

    3. Taylor expanded around 0 13.4

      \[\leadsto \frac{\sqrt{\color{blue}{{b}^{2} - 3 \cdot \left(a \cdot c\right)}} - b}{3 \cdot a}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt13.6

      \[\leadsto \frac{\sqrt{{b}^{2} - \color{blue}{\left(\sqrt[3]{3 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{3 \cdot \left(a \cdot c\right)}\right) \cdot \sqrt[3]{3 \cdot \left(a \cdot c\right)}}} - b}{3 \cdot a}\]
    6. Using strategy rm

    7. Applied cbrt-prod13.5

      \[\leadsto \frac{\sqrt{{b}^{2} - \left(\sqrt[3]{3 \cdot \left(a \cdot c\right)} \cdot \color{blue}{\left(\sqrt[3]{3} \cdot \sqrt[3]{a \cdot c}\right)}\right) \cdot \sqrt[3]{3 \cdot \left(a \cdot c\right)}} - b}{3 \cdot a}\]

    if 3.5034007549662594e-57 < b

    1. Initial program 53.9

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

    2. Simplified53.9

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

    3. Taylor expanded around inf 19.8

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification15.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.2724541866372811 \cdot 10^{165}:\\ \;\;\;\;\frac{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \le 3.50340075496625935 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{{b}^{2} - \left(\sqrt[3]{3 \cdot \left(a \cdot c\right)} \cdot \left(\sqrt[3]{3} \cdot \sqrt[3]{a \cdot c}\right)\right) \cdot \sqrt[3]{3 \cdot \left(a \cdot c\right)}} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))