Average Error: 28.5 → 16.2
Time: 6.8s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 904.955392750566375:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - 3 \cdot \left(a \cdot c\right)\right) - {b}^{2}}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5}{3} \cdot \frac{\frac{a \cdot c}{b}}{a}\\ \end{array}\]
\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 904.955392750566375:\\
\;\;\;\;\frac{\frac{\left({b}^{2} - 3 \cdot \left(a \cdot c\right)\right) - {b}^{2}}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r78107 = b;
        double r78108 = -r78107;
        double r78109 = r78107 * r78107;
        double r78110 = 3.0;
        double r78111 = a;
        double r78112 = r78110 * r78111;
        double r78113 = c;
        double r78114 = r78112 * r78113;
        double r78115 = r78109 - r78114;
        double r78116 = sqrt(r78115);
        double r78117 = r78108 + r78116;
        double r78118 = r78117 / r78112;
        return r78118;
}
↓
double f(double a, double b, double c) {
        double r78119 = b;
        double r78120 = 904.9553927505664;
        bool r78121 = r78119 <= r78120;
        double r78122 = 2.0;
        double r78123 = pow(r78119, r78122);
        double r78124 = 3.0;
        double r78125 = a;
        double r78126 = c;
        double r78127 = r78125 * r78126;
        double r78128 = r78124 * r78127;
        double r78129 = r78123 - r78128;
        double r78130 = r78129 - r78123;
        double r78131 = r78119 * r78119;
        double r78132 = r78124 * r78125;
        double r78133 = r78132 * r78126;
        double r78134 = r78131 - r78133;
        double r78135 = sqrt(r78134);
        double r78136 = r78135 + r78119;
        double r78137 = r78130 / r78136;
        double r78138 = r78137 / r78132;
        double r78139 = -1.5;
        double r78140 = r78139 / r78124;
        double r78141 = r78127 / r78119;
        double r78142 = r78141 / r78125;
        double r78143 = r78140 * r78142;
        double r78144 = r78121 ? r78138 : r78143;
        return r78144;
}

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 2 regimes

  2. if b < 904.9553927505664

    1. Initial program 17.2

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

    2. Simplified17.2

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

    4. Applied flip--17.2

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

    5. Simplified16.1

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

    if 904.9553927505664 < b

    1. Initial program 36.2

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

    2. Simplified36.2

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

    3. Taylor expanded around inf 16.4

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]
    4. Using strategy rm

    5. Applied times-frac16.3

      \[\leadsto \color{blue}{\frac{-1.5}{3} \cdot \frac{\frac{a \cdot c}{b}}{a}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 904.955392750566375:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - 3 \cdot \left(a \cdot c\right)\right) - {b}^{2}}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5}{3} \cdot \frac{\frac{a \cdot c}{b}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))