Average Error: 44.0 → 11.0
Time: 14.7s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.005465639874354823:\\ \;\;\;\;\frac{\frac{\left(\left(a \cdot c\right) \cdot -3 + b \cdot b\right) \cdot \sqrt{\left(a \cdot c\right) \cdot -3 + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(\left(a \cdot c\right) \cdot -3 + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{\left(a \cdot c\right) \cdot -3 + b \cdot b}\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 0.005465639874354823:\\
\;\;\;\;\frac{\frac{\left(\left(a \cdot c\right) \cdot -3 + b \cdot b\right) \cdot \sqrt{\left(a \cdot c\right) \cdot -3 + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(\left(a \cdot c\right) \cdot -3 + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{\left(a \cdot c\right) \cdot -3 + 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 __attribute__((unused)) d) {
        double r1818987 = b;
        double r1818988 = -r1818987;
        double r1818989 = r1818987 * r1818987;
        double r1818990 = 3.0;
        double r1818991 = a;
        double r1818992 = r1818990 * r1818991;
        double r1818993 = c;
        double r1818994 = r1818992 * r1818993;
        double r1818995 = r1818989 - r1818994;
        double r1818996 = sqrt(r1818995);
        double r1818997 = r1818988 + r1818996;
        double r1818998 = r1818997 / r1818992;
        return r1818998;
}
↓
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r1818999 = b;
        double r1819000 = 0.005465639874354823;
        bool r1819001 = r1818999 <= r1819000;
        double r1819002 = a;
        double r1819003 = c;
        double r1819004 = r1819002 * r1819003;
        double r1819005 = -3.0;
        double r1819006 = r1819004 * r1819005;
        double r1819007 = r1818999 * r1818999;
        double r1819008 = r1819006 + r1819007;
        double r1819009 = sqrt(r1819008);
        double r1819010 = r1819008 * r1819009;
        double r1819011 = r1818999 * r1819007;
        double r1819012 = r1819010 - r1819011;
        double r1819013 = r1818999 * r1819009;
        double r1819014 = r1819007 + r1819013;
        double r1819015 = r1819008 + r1819014;
        double r1819016 = r1819012 / r1819015;
        double r1819017 = 3.0;
        double r1819018 = r1819002 * r1819017;
        double r1819019 = r1819016 / r1819018;
        double r1819020 = -0.5;
        double r1819021 = r1819003 / r1818999;
        double r1819022 = r1819020 * r1819021;
        double r1819023 = r1819001 ? r1819019 : r1819022;
        return r1819023;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if b < 0.005465639874354823

    1. Initial program 21.3

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

    2. Simplified21.3

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

    4. Applied flip3--21.4

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

    5. Simplified20.7

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

    6. Simplified20.7

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

    if 0.005465639874354823 < b

    1. Initial program 46.6

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

    2. Simplified46.6

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

    3. Taylor expanded around inf 9.9

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

  4. Final simplification11.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.005465639874354823:\\ \;\;\;\;\frac{\frac{\left(\left(a \cdot c\right) \cdot -3 + b \cdot b\right) \cdot \sqrt{\left(a \cdot c\right) \cdot -3 + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(\left(a \cdot c\right) \cdot -3 + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{\left(a \cdot c\right) \cdot -3 + b \cdot b}\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(FPCore (a b c d)
  :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)))