Average Error: 28.7 → 16.6
Time: 13.9s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 2892.1913455639924:\\ \;\;\;\;\frac{\frac{\left(\left(-3 \cdot a\right) \cdot c + b \cdot b\right) \cdot \sqrt{\left(-3 \cdot a\right) \cdot c + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(\left(-3 \cdot a\right) \cdot c + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{\left(-3 \cdot a\right) \cdot c + b \cdot b}\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 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 \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r1760407 = b;
        double r1760408 = -r1760407;
        double r1760409 = r1760407 * r1760407;
        double r1760410 = 3.0;
        double r1760411 = a;
        double r1760412 = r1760410 * r1760411;
        double r1760413 = c;
        double r1760414 = r1760412 * r1760413;
        double r1760415 = r1760409 - r1760414;
        double r1760416 = sqrt(r1760415);
        double r1760417 = r1760408 + r1760416;
        double r1760418 = r1760417 / r1760412;
        return r1760418;
}
↓
double f(double a, double b, double c) {
        double r1760419 = b;
        double r1760420 = 2892.1913455639924;
        bool r1760421 = r1760419 <= r1760420;
        double r1760422 = -3.0;
        double r1760423 = a;
        double r1760424 = r1760422 * r1760423;
        double r1760425 = c;
        double r1760426 = r1760424 * r1760425;
        double r1760427 = r1760419 * r1760419;
        double r1760428 = r1760426 + r1760427;
        double r1760429 = sqrt(r1760428);
        double r1760430 = r1760428 * r1760429;
        double r1760431 = r1760419 * r1760427;
        double r1760432 = r1760430 - r1760431;
        double r1760433 = r1760419 * r1760429;
        double r1760434 = r1760427 + r1760433;
        double r1760435 = r1760428 + r1760434;
        double r1760436 = r1760432 / r1760435;
        double r1760437 = 3.0;
        double r1760438 = r1760423 * r1760437;
        double r1760439 = r1760436 / r1760438;
        double r1760440 = -0.5;
        double r1760441 = r1760425 / r1760419;
        double r1760442 = r1760440 * r1760441;
        double r1760443 = r1760421 ? r1760439 : r1760442;
        return r1760443;
}

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

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

    6. Simplified18.0

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

    if 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. Taylor expanded around 0 15.5

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

  4. Final simplification16.6

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

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)))