Average Error: 43.6 → 11.3
Time: 49.9s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.0015085102264598175:\\ \;\;\;\;\frac{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(3 \cdot a\right) \cdot \left(\left(b \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} + b \cdot b\right) + \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 0.0015085102264598175:\\
\;\;\;\;\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(3 \cdot a\right) \cdot \left(\left(b \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} + b \cdot b\right) + \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}\right)}\\

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

\end{array}
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r14415263 = b;
        double r14415264 = -r14415263;
        double r14415265 = r14415263 * r14415263;
        double r14415266 = 3.0;
        double r14415267 = a;
        double r14415268 = r14415266 * r14415267;
        double r14415269 = c;
        double r14415270 = r14415268 * r14415269;
        double r14415271 = r14415265 - r14415270;
        double r14415272 = sqrt(r14415271);
        double r14415273 = r14415264 + r14415272;
        double r14415274 = r14415273 / r14415268;
        return r14415274;
}
↓
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r14415275 = b;
        double r14415276 = 0.0015085102264598175;
        bool r14415277 = r14415275 <= r14415276;
        double r14415278 = r14415275 * r14415275;
        double r14415279 = c;
        double r14415280 = 3.0;
        double r14415281 = a;
        double r14415282 = r14415280 * r14415281;
        double r14415283 = r14415279 * r14415282;
        double r14415284 = r14415278 - r14415283;
        double r14415285 = sqrt(r14415284);
        double r14415286 = r14415284 * r14415285;
        double r14415287 = r14415278 * r14415275;
        double r14415288 = r14415286 - r14415287;
        double r14415289 = r14415275 * r14415285;
        double r14415290 = r14415289 + r14415278;
        double r14415291 = r14415285 * r14415285;
        double r14415292 = r14415290 + r14415291;
        double r14415293 = r14415282 * r14415292;
        double r14415294 = r14415288 / r14415293;
        double r14415295 = -0.5;
        double r14415296 = r14415279 / r14415275;
        double r14415297 = r14415295 * r14415296;
        double r14415298 = r14415277 ? r14415294 : r14415297;
        return r14415298;
}

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 < 0.0015085102264598175

    1. Initial program 20.6

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

    2. Simplified20.6

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

    4. Applied flip3--20.7

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

    5. Applied associate-/l/20.7

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

    6. Simplified20.0

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

    if 0.0015085102264598175 < b

    1. Initial program 45.9

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

    2. Simplified45.9

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

    3. Taylor expanded around inf 10.5

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

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.0015085102264598175:\\ \;\;\;\;\frac{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(3 \cdot a\right) \cdot \left(\left(b \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} + b \cdot b\right) + \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 
(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)))