Average Error: 44.1 → 11.2
Time: 15.3s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.10455593216828488:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - a \cdot \left(3 \cdot c\right)\right) \cdot \sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - a \cdot \left(3 \cdot c\right)\right) + \left(b \cdot \sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} + b \cdot b\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 0.10455593216828488:\\
\;\;\;\;\frac{\frac{\left(b \cdot b - a \cdot \left(3 \cdot c\right)\right) \cdot \sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - a \cdot \left(3 \cdot c\right)\right) + \left(b \cdot \sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} + b \cdot b\right)}}{a \cdot 3}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r4769122 = b;
        double r4769123 = -r4769122;
        double r4769124 = r4769122 * r4769122;
        double r4769125 = 3.0;
        double r4769126 = a;
        double r4769127 = r4769125 * r4769126;
        double r4769128 = c;
        double r4769129 = r4769127 * r4769128;
        double r4769130 = r4769124 - r4769129;
        double r4769131 = sqrt(r4769130);
        double r4769132 = r4769123 + r4769131;
        double r4769133 = r4769132 / r4769127;
        return r4769133;
}
↓
double f(double a, double b, double c) {
        double r4769134 = b;
        double r4769135 = 0.10455593216828488;
        bool r4769136 = r4769134 <= r4769135;
        double r4769137 = r4769134 * r4769134;
        double r4769138 = a;
        double r4769139 = 3.0;
        double r4769140 = c;
        double r4769141 = r4769139 * r4769140;
        double r4769142 = r4769138 * r4769141;
        double r4769143 = r4769137 - r4769142;
        double r4769144 = sqrt(r4769143);
        double r4769145 = r4769143 * r4769144;
        double r4769146 = r4769137 * r4769134;
        double r4769147 = r4769145 - r4769146;
        double r4769148 = r4769134 * r4769144;
        double r4769149 = r4769148 + r4769137;
        double r4769150 = r4769143 + r4769149;
        double r4769151 = r4769147 / r4769150;
        double r4769152 = r4769138 * r4769139;
        double r4769153 = r4769151 / r4769152;
        double r4769154 = -0.5;
        double r4769155 = r4769140 / r4769134;
        double r4769156 = r4769154 * r4769155;
        double r4769157 = r4769136 ? r4769153 : r4769156;
        return r4769157;
}

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 < 0.10455593216828488

    1. Initial program 23.1

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

    2. Simplified23.1

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

    4. Applied flip3--23.2

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

    5. Simplified22.6

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

    6. Simplified22.6

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

    if 0.10455593216828488 < b

    1. Initial program 47.3

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

    2. Simplified47.3

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

    3. Taylor expanded around inf 9.5

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.10455593216828488:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - a \cdot \left(3 \cdot c\right)\right) \cdot \sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - a \cdot \left(3 \cdot c\right)\right) + \left(b \cdot \sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} + b \cdot b\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))