Average Error: 28.7 → 16.6
Time: 33.2s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 2892.1913455639924:\\ \;\;\;\;\frac{\frac{\left(\left(-3 \cdot c\right) \cdot a + b \cdot b\right) \cdot \sqrt{\left(-3 \cdot c\right) \cdot a + b \cdot b} - b \cdot \left(b \cdot b\right)}{b \cdot \sqrt{\left(-3 \cdot c\right) \cdot a + b \cdot b} + \left(b \cdot b + \left(\left(-3 \cdot c\right) \cdot a + b \cdot b\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b}{c \cdot \frac{-1}{2}}}\\ \end{array}\]
\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 2892.1913455639924:\\
\;\;\;\;\frac{\frac{\left(\left(-3 \cdot c\right) \cdot a + b \cdot b\right) \cdot \sqrt{\left(-3 \cdot c\right) \cdot a + b \cdot b} - b \cdot \left(b \cdot b\right)}{b \cdot \sqrt{\left(-3 \cdot c\right) \cdot a + b \cdot b} + \left(b \cdot b + \left(\left(-3 \cdot c\right) \cdot a + b \cdot b\right)\right)}}{3 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r4094755 = b;
        double r4094756 = -r4094755;
        double r4094757 = r4094755 * r4094755;
        double r4094758 = 3.0;
        double r4094759 = a;
        double r4094760 = r4094758 * r4094759;
        double r4094761 = c;
        double r4094762 = r4094760 * r4094761;
        double r4094763 = r4094757 - r4094762;
        double r4094764 = sqrt(r4094763);
        double r4094765 = r4094756 + r4094764;
        double r4094766 = r4094765 / r4094760;
        return r4094766;
}
↓
double f(double a, double b, double c) {
        double r4094767 = b;
        double r4094768 = 2892.1913455639924;
        bool r4094769 = r4094767 <= r4094768;
        double r4094770 = -3.0;
        double r4094771 = c;
        double r4094772 = r4094770 * r4094771;
        double r4094773 = a;
        double r4094774 = r4094772 * r4094773;
        double r4094775 = r4094767 * r4094767;
        double r4094776 = r4094774 + r4094775;
        double r4094777 = sqrt(r4094776);
        double r4094778 = r4094776 * r4094777;
        double r4094779 = r4094767 * r4094775;
        double r4094780 = r4094778 - r4094779;
        double r4094781 = r4094767 * r4094777;
        double r4094782 = r4094775 + r4094776;
        double r4094783 = r4094781 + r4094782;
        double r4094784 = r4094780 / r4094783;
        double r4094785 = 3.0;
        double r4094786 = r4094785 * r4094773;
        double r4094787 = r4094784 / r4094786;
        double r4094788 = 1.0;
        double r4094789 = -0.5;
        double r4094790 = r4094771 * r4094789;
        double r4094791 = r4094767 / r4094790;
        double r4094792 = r4094788 / r4094791;
        double r4094793 = r4094769 ? r4094787 : r4094792;
        return r4094793;
}

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 < 2892.1913455639924

    1. Initial program 18.6

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

    2. Simplified18.6

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

    4. Applied flip3--18.7

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

    5. Simplified18.0

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

    6. Simplified18.0

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

    if 2892.1913455639924 < b

    1. Initial program 37.1

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

    2. Simplified37.1

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

    3. Taylor expanded around inf 15.6

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

    5. Applied clear-num15.6

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

    6. Simplified15.5

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

  4. Final simplification16.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2892.1913455639924:\\ \;\;\;\;\frac{\frac{\left(\left(-3 \cdot c\right) \cdot a + b \cdot b\right) \cdot \sqrt{\left(-3 \cdot c\right) \cdot a + b \cdot b} - b \cdot \left(b \cdot b\right)}{b \cdot \sqrt{\left(-3 \cdot c\right) \cdot a + b \cdot b} + \left(b \cdot b + \left(\left(-3 \cdot c\right) \cdot a + b \cdot b\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b}{c \cdot \frac{-1}{2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(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)))