Average Error: 28.7 → 16.6
Time: 14.3s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 5152.464935290847:\\ \;\;\;\;\frac{\frac{\left(\left(a \cdot c\right) \cdot -3 + b \cdot b\right) \cdot \sqrt{\left(a \cdot c\right) \cdot -3 + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(\left(a \cdot c\right) \cdot -3 + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{\left(a \cdot c\right) \cdot -3 + b \cdot b}\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a \cdot b}{\frac{a \cdot c}{-2}}}\\ \end{array}\]
\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 5152.464935290847:\\
\;\;\;\;\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}{\frac{a \cdot b}{\frac{a \cdot c}{-2}}}\\

\end{array}
double f(double a, double b, double c) {
        double r2270837 = b;
        double r2270838 = -r2270837;
        double r2270839 = r2270837 * r2270837;
        double r2270840 = 3.0;
        double r2270841 = a;
        double r2270842 = r2270840 * r2270841;
        double r2270843 = c;
        double r2270844 = r2270842 * r2270843;
        double r2270845 = r2270839 - r2270844;
        double r2270846 = sqrt(r2270845);
        double r2270847 = r2270838 + r2270846;
        double r2270848 = r2270847 / r2270842;
        return r2270848;
}
↓
double f(double a, double b, double c) {
        double r2270849 = b;
        double r2270850 = 5152.464935290847;
        bool r2270851 = r2270849 <= r2270850;
        double r2270852 = a;
        double r2270853 = c;
        double r2270854 = r2270852 * r2270853;
        double r2270855 = -3.0;
        double r2270856 = r2270854 * r2270855;
        double r2270857 = r2270849 * r2270849;
        double r2270858 = r2270856 + r2270857;
        double r2270859 = sqrt(r2270858);
        double r2270860 = r2270858 * r2270859;
        double r2270861 = r2270849 * r2270857;
        double r2270862 = r2270860 - r2270861;
        double r2270863 = r2270849 * r2270859;
        double r2270864 = r2270857 + r2270863;
        double r2270865 = r2270858 + r2270864;
        double r2270866 = r2270862 / r2270865;
        double r2270867 = 3.0;
        double r2270868 = r2270852 * r2270867;
        double r2270869 = r2270866 / r2270868;
        double r2270870 = 1.0;
        double r2270871 = r2270852 * r2270849;
        double r2270872 = -2.0;
        double r2270873 = r2270854 / r2270872;
        double r2270874 = r2270871 / r2270873;
        double r2270875 = r2270870 / r2270874;
        double r2270876 = r2270851 ? r2270869 : r2270875;
        return r2270876;
}

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 < 5152.464935290847

    1. Initial program 18.9

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

    2. Simplified18.9

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

    4. Applied flip3--19.1

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

    5. Simplified18.3

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

    6. Simplified18.3

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

    if 5152.464935290847 < b

    1. Initial program 38.2

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

    2. Simplified38.2

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

    3. Taylor expanded around inf 14.9

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

    5. Applied associate-*r/14.9

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

    6. Applied associate-/l/14.9

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

    8. Applied clear-num15.0

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(3 \cdot a\right) \cdot b}{\frac{-3}{2} \cdot \left(a \cdot c\right)}}}\]

    9. Simplified14.9

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

  4. Final simplification16.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 5152.464935290847:\\ \;\;\;\;\frac{\frac{\left(\left(a \cdot c\right) \cdot -3 + b \cdot b\right) \cdot \sqrt{\left(a \cdot c\right) \cdot -3 + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(\left(a \cdot c\right) \cdot -3 + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{\left(a \cdot c\right) \cdot -3 + b \cdot b}\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a \cdot b}{\frac{a \cdot c}{-2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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)))