Average Error: 28.7 → 16.3
Time: 15.9s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 2495.5039318207096:\\ \;\;\;\;\frac{\frac{\left(\left(-3 \cdot a\right) \cdot c + b \cdot b\right) \cdot \sqrt{\left(-3 \cdot a\right) \cdot c + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(\left(-3 \cdot a\right) \cdot c + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{\left(-3 \cdot a\right) \cdot c + b \cdot b}\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \left(\frac{\frac{1}{b}}{a} \cdot \left(c \cdot a\right)\right)\\ \end{array}\]
\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 2495.5039318207096:\\
\;\;\;\;\frac{\frac{\left(\left(-3 \cdot a\right) \cdot c + b \cdot b\right) \cdot \sqrt{\left(-3 \cdot a\right) \cdot c + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(\left(-3 \cdot a\right) \cdot c + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{\left(-3 \cdot a\right) \cdot c + b \cdot b}\right)}}{a \cdot 3}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{2} \cdot \left(\frac{\frac{1}{b}}{a} \cdot \left(c \cdot a\right)\right)\\

\end{array}
double f(double a, double b, double c) {
        double r1316863 = b;
        double r1316864 = -r1316863;
        double r1316865 = r1316863 * r1316863;
        double r1316866 = 3.0;
        double r1316867 = a;
        double r1316868 = r1316866 * r1316867;
        double r1316869 = c;
        double r1316870 = r1316868 * r1316869;
        double r1316871 = r1316865 - r1316870;
        double r1316872 = sqrt(r1316871);
        double r1316873 = r1316864 + r1316872;
        double r1316874 = r1316873 / r1316868;
        return r1316874;
}
↓
double f(double a, double b, double c) {
        double r1316875 = b;
        double r1316876 = 2495.5039318207096;
        bool r1316877 = r1316875 <= r1316876;
        double r1316878 = -3.0;
        double r1316879 = a;
        double r1316880 = r1316878 * r1316879;
        double r1316881 = c;
        double r1316882 = r1316880 * r1316881;
        double r1316883 = r1316875 * r1316875;
        double r1316884 = r1316882 + r1316883;
        double r1316885 = sqrt(r1316884);
        double r1316886 = r1316884 * r1316885;
        double r1316887 = r1316875 * r1316883;
        double r1316888 = r1316886 - r1316887;
        double r1316889 = r1316875 * r1316885;
        double r1316890 = r1316883 + r1316889;
        double r1316891 = r1316884 + r1316890;
        double r1316892 = r1316888 / r1316891;
        double r1316893 = 3.0;
        double r1316894 = r1316879 * r1316893;
        double r1316895 = r1316892 / r1316894;
        double r1316896 = -0.5;
        double r1316897 = 1.0;
        double r1316898 = r1316897 / r1316875;
        double r1316899 = r1316898 / r1316879;
        double r1316900 = r1316881 * r1316879;
        double r1316901 = r1316899 * r1316900;
        double r1316902 = r1316896 * r1316901;
        double r1316903 = r1316877 ? r1316895 : r1316902;
        return r1316903;
}

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 < 2495.5039318207096

    1. Initial program 18.0

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

    2. Simplified18.0

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

    4. Applied flip3--18.1

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

    5. Simplified17.4

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

    6. Simplified17.4

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

    if 2495.5039318207096 < b

    1. Initial program 37.5

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

    2. Simplified37.5

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

    3. Taylor expanded around inf 15.5

      \[\leadsto \frac{\color{blue}{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]
    4. Using strategy rm

    5. Applied times-frac15.4

      \[\leadsto \color{blue}{\frac{\frac{-3}{2}}{3} \cdot \frac{\frac{a \cdot c}{b}}{a}}\]

    6. Simplified15.4

      \[\leadsto \color{blue}{\frac{-1}{2}} \cdot \frac{\frac{a \cdot c}{b}}{a}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity15.4

      \[\leadsto \frac{-1}{2} \cdot \frac{\frac{a \cdot c}{b}}{\color{blue}{1 \cdot a}}\]

    9. Applied div-inv15.5

      \[\leadsto \frac{-1}{2} \cdot \frac{\color{blue}{\left(a \cdot c\right) \cdot \frac{1}{b}}}{1 \cdot a}\]

    10. Applied times-frac15.5

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{\left(\frac{a \cdot c}{1} \cdot \frac{\frac{1}{b}}{a}\right)}\]

    11. Simplified15.5

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

  4. Final simplification16.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2495.5039318207096:\\ \;\;\;\;\frac{\frac{\left(\left(-3 \cdot a\right) \cdot c + b \cdot b\right) \cdot \sqrt{\left(-3 \cdot a\right) \cdot c + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(\left(-3 \cdot a\right) \cdot c + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{\left(-3 \cdot a\right) \cdot c + b \cdot b}\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \left(\frac{\frac{1}{b}}{a} \cdot \left(c \cdot a\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :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)))