Average Error: 44.4 → 11.3
Time: 32.0s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 3.826212081602258 \cdot 10^{-05}:\\ \;\;\;\;\frac{\frac{\left(b \cdot b + -3 \cdot \left(a \cdot c\right)\right) \cdot \sqrt{b \cdot b + -3 \cdot \left(a \cdot c\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b + -3 \cdot \left(a \cdot c\right)\right) + \left(b \cdot \sqrt{b \cdot b + -3 \cdot \left(a \cdot c\right)} + b \cdot b\right)}}{a \cdot 3}\\ \mathbf{elif}\;b \le 0.015132399096455612:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\ \mathbf{elif}\;b \le 1.3718256577401997:\\ \;\;\;\;\frac{\frac{\left(b \cdot b + -3 \cdot \left(a \cdot c\right)\right) \cdot \sqrt{b \cdot b + -3 \cdot \left(a \cdot c\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b + -3 \cdot \left(a \cdot c\right)\right) + \left(b \cdot \sqrt{b \cdot b + -3 \cdot \left(a \cdot c\right)} + b \cdot b\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\ \end{array}\]
\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 3.826212081602258 \cdot 10^{-05}:\\
\;\;\;\;\frac{\frac{\left(b \cdot b + -3 \cdot \left(a \cdot c\right)\right) \cdot \sqrt{b \cdot b + -3 \cdot \left(a \cdot c\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b + -3 \cdot \left(a \cdot c\right)\right) + \left(b \cdot \sqrt{b \cdot b + -3 \cdot \left(a \cdot c\right)} + b \cdot b\right)}}{a \cdot 3}\\

\mathbf{elif}\;b \le 0.015132399096455612:\\
\;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\

\mathbf{elif}\;b \le 1.3718256577401997:\\
\;\;\;\;\frac{\frac{\left(b \cdot b + -3 \cdot \left(a \cdot c\right)\right) \cdot \sqrt{b \cdot b + -3 \cdot \left(a \cdot c\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b + -3 \cdot \left(a \cdot c\right)\right) + \left(b \cdot \sqrt{b \cdot b + -3 \cdot \left(a \cdot c\right)} + b \cdot b\right)}}{a \cdot 3}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3946536 = b;
        double r3946537 = -r3946536;
        double r3946538 = r3946536 * r3946536;
        double r3946539 = 3.0;
        double r3946540 = a;
        double r3946541 = r3946539 * r3946540;
        double r3946542 = c;
        double r3946543 = r3946541 * r3946542;
        double r3946544 = r3946538 - r3946543;
        double r3946545 = sqrt(r3946544);
        double r3946546 = r3946537 + r3946545;
        double r3946547 = r3946546 / r3946541;
        return r3946547;
}
↓
double f(double a, double b, double c) {
        double r3946548 = b;
        double r3946549 = 3.826212081602258e-05;
        bool r3946550 = r3946548 <= r3946549;
        double r3946551 = r3946548 * r3946548;
        double r3946552 = -3.0;
        double r3946553 = a;
        double r3946554 = c;
        double r3946555 = r3946553 * r3946554;
        double r3946556 = r3946552 * r3946555;
        double r3946557 = r3946551 + r3946556;
        double r3946558 = sqrt(r3946557);
        double r3946559 = r3946557 * r3946558;
        double r3946560 = r3946551 * r3946548;
        double r3946561 = r3946559 - r3946560;
        double r3946562 = r3946548 * r3946558;
        double r3946563 = r3946562 + r3946551;
        double r3946564 = r3946557 + r3946563;
        double r3946565 = r3946561 / r3946564;
        double r3946566 = 3.0;
        double r3946567 = r3946553 * r3946566;
        double r3946568 = r3946565 / r3946567;
        double r3946569 = 0.015132399096455612;
        bool r3946570 = r3946548 <= r3946569;
        double r3946571 = r3946554 / r3946548;
        double r3946572 = -0.5;
        double r3946573 = r3946571 * r3946572;
        double r3946574 = 1.3718256577401997;
        bool r3946575 = r3946548 <= r3946574;
        double r3946576 = r3946575 ? r3946568 : r3946573;
        double r3946577 = r3946570 ? r3946573 : r3946576;
        double r3946578 = r3946550 ? r3946568 : r3946577;
        return r3946578;
}

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 < 3.826212081602258e-05 or 0.015132399096455612 < b < 1.3718256577401997

    1. Initial program 24.8

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

    2. Simplified24.8

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

    4. Applied flip3--24.8

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

    5. Simplified24.1

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

    6. Simplified24.1

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

    if 3.826212081602258e-05 < b < 0.015132399096455612 or 1.3718256577401997 < b

    1. Initial program 46.7

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

    2. Simplified46.7

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

    3. Taylor expanded around inf 9.8

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

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 3.826212081602258 \cdot 10^{-05}:\\ \;\;\;\;\frac{\frac{\left(b \cdot b + -3 \cdot \left(a \cdot c\right)\right) \cdot \sqrt{b \cdot b + -3 \cdot \left(a \cdot c\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b + -3 \cdot \left(a \cdot c\right)\right) + \left(b \cdot \sqrt{b \cdot b + -3 \cdot \left(a \cdot c\right)} + b \cdot b\right)}}{a \cdot 3}\\ \mathbf{elif}\;b \le 0.015132399096455612:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\ \mathbf{elif}\;b \le 1.3718256577401997:\\ \;\;\;\;\frac{\frac{\left(b \cdot b + -3 \cdot \left(a \cdot c\right)\right) \cdot \sqrt{b \cdot b + -3 \cdot \left(a \cdot c\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b + -3 \cdot \left(a \cdot c\right)\right) + \left(b \cdot \sqrt{b \cdot b + -3 \cdot \left(a \cdot c\right)} + b \cdot b\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (a b c)
  :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)))