Average Error: 28.7 → 16.3
Time: 14.4s
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 r1701669 = b;
        double r1701670 = -r1701669;
        double r1701671 = r1701669 * r1701669;
        double r1701672 = 3.0;
        double r1701673 = a;
        double r1701674 = r1701672 * r1701673;
        double r1701675 = c;
        double r1701676 = r1701674 * r1701675;
        double r1701677 = r1701671 - r1701676;
        double r1701678 = sqrt(r1701677);
        double r1701679 = r1701670 + r1701678;
        double r1701680 = r1701679 / r1701674;
        return r1701680;
}
↓
double f(double a, double b, double c) {
        double r1701681 = b;
        double r1701682 = 2495.5039318207096;
        bool r1701683 = r1701681 <= r1701682;
        double r1701684 = -3.0;
        double r1701685 = a;
        double r1701686 = r1701684 * r1701685;
        double r1701687 = c;
        double r1701688 = r1701686 * r1701687;
        double r1701689 = r1701681 * r1701681;
        double r1701690 = r1701688 + r1701689;
        double r1701691 = sqrt(r1701690);
        double r1701692 = r1701690 * r1701691;
        double r1701693 = r1701681 * r1701689;
        double r1701694 = r1701692 - r1701693;
        double r1701695 = r1701681 * r1701691;
        double r1701696 = r1701689 + r1701695;
        double r1701697 = r1701690 + r1701696;
        double r1701698 = r1701694 / r1701697;
        double r1701699 = 3.0;
        double r1701700 = r1701685 * r1701699;
        double r1701701 = r1701698 / r1701700;
        double r1701702 = -0.5;
        double r1701703 = 1.0;
        double r1701704 = r1701703 / r1701681;
        double r1701705 = r1701704 / r1701685;
        double r1701706 = r1701687 * r1701685;
        double r1701707 = r1701705 * r1701706;
        double r1701708 = r1701702 * r1701707;
        double r1701709 = r1701683 ? r1701701 : r1701708;
        return r1701709;
}

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)))