Average Error: 43.8 → 11.1
Time: 6.6s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 1.7936624356974993 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right) - {b}^{2}}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 1.7936624356974993 \cdot 10^{-4}:\\
\;\;\;\;\frac{\frac{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right) - {b}^{2}}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;-0.5 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r158474 = b;
        double r158475 = -r158474;
        double r158476 = r158474 * r158474;
        double r158477 = 3.0;
        double r158478 = a;
        double r158479 = r158477 * r158478;
        double r158480 = c;
        double r158481 = r158479 * r158480;
        double r158482 = r158476 - r158481;
        double r158483 = sqrt(r158482);
        double r158484 = r158475 + r158483;
        double r158485 = r158484 / r158479;
        return r158485;
}
↓
double f(double a, double b, double c) {
        double r158486 = b;
        double r158487 = 0.00017936624356974993;
        bool r158488 = r158486 <= r158487;
        double r158489 = r158486 * r158486;
        double r158490 = 3.0;
        double r158491 = a;
        double r158492 = r158490 * r158491;
        double r158493 = c;
        double r158494 = r158492 * r158493;
        double r158495 = r158489 - r158494;
        double r158496 = 2.0;
        double r158497 = pow(r158486, r158496);
        double r158498 = r158495 - r158497;
        double r158499 = sqrt(r158495);
        double r158500 = r158499 + r158486;
        double r158501 = r158498 / r158500;
        double r158502 = r158501 / r158492;
        double r158503 = -0.5;
        double r158504 = r158493 / r158486;
        double r158505 = r158503 * r158504;
        double r158506 = r158488 ? r158502 : r158505;
        return r158506;
}

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 < 0.00017936624356974993

    1. Initial program 18.4

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

    2. Simplified18.4

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

    4. Applied flip--18.4

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

    5. Simplified17.4

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

    if 0.00017936624356974993 < b

    1. Initial program 45.7

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

    2. Simplified45.7

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

    3. Taylor expanded around inf 10.6

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1.7936624356974993 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right) - {b}^{2}}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e+15) (< 1.11022e-16 b 9.0072e+15) (< 1.11022e-16 c 9.0072e+15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))