Average Error: 44.0 → 11.0
Time: 1.4m
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.005465639874354823:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b + \left(b \cdot b - c \cdot \left(3 \cdot a\right)\right)\right) + b \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 0.005465639874354823:\\
\;\;\;\;\frac{\frac{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b + \left(b \cdot b - c \cdot \left(3 \cdot a\right)\right)\right) + b \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}}}{3 \cdot a}\\

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

\end{array}
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r22790261 = b;
        double r22790262 = -r22790261;
        double r22790263 = r22790261 * r22790261;
        double r22790264 = 3.0;
        double r22790265 = a;
        double r22790266 = r22790264 * r22790265;
        double r22790267 = c;
        double r22790268 = r22790266 * r22790267;
        double r22790269 = r22790263 - r22790268;
        double r22790270 = sqrt(r22790269);
        double r22790271 = r22790262 + r22790270;
        double r22790272 = r22790271 / r22790266;
        return r22790272;
}
↓
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r22790273 = b;
        double r22790274 = 0.005465639874354823;
        bool r22790275 = r22790273 <= r22790274;
        double r22790276 = r22790273 * r22790273;
        double r22790277 = c;
        double r22790278 = 3.0;
        double r22790279 = a;
        double r22790280 = r22790278 * r22790279;
        double r22790281 = r22790277 * r22790280;
        double r22790282 = r22790276 - r22790281;
        double r22790283 = sqrt(r22790282);
        double r22790284 = r22790282 * r22790283;
        double r22790285 = r22790276 * r22790273;
        double r22790286 = r22790284 - r22790285;
        double r22790287 = r22790276 + r22790282;
        double r22790288 = r22790273 * r22790283;
        double r22790289 = r22790287 + r22790288;
        double r22790290 = r22790286 / r22790289;
        double r22790291 = r22790290 / r22790280;
        double r22790292 = -0.5;
        double r22790293 = r22790277 / r22790273;
        double r22790294 = r22790292 * r22790293;
        double r22790295 = r22790275 ? r22790291 : r22790294;
        return r22790295;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if b < 0.005465639874354823

    1. Initial program 21.3

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

    2. Simplified21.3

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

    4. Applied flip3--21.4

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

    5. Simplified20.7

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

    6. Simplified20.7

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

    if 0.005465639874354823 < b

    1. Initial program 46.6

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

    2. Simplified46.6

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

    3. Taylor expanded around inf 9.9

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

  4. Final simplification11.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.005465639874354823:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b + \left(b \cdot b - c \cdot \left(3 \cdot a\right)\right)\right) + b \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(FPCore (a b c d)
  :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)))