Average Error: 44.0 → 11.4
Time: 17.3s
Precision: 64
\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 6.961131476357276728544534868600712762543 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - c \cdot \left(a \cdot 3\right)\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - c \cdot \left(a \cdot 3\right)\right) + \left(b \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} + b \cdot b\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 6.961131476357276728544534868600712762543 \cdot 10^{-8}:\\
\;\;\;\;\frac{\frac{\left(b \cdot b - c \cdot \left(a \cdot 3\right)\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - c \cdot \left(a \cdot 3\right)\right) + \left(b \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} + b \cdot b\right)}}{a \cdot 3}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2806073 = b;
        double r2806074 = -r2806073;
        double r2806075 = r2806073 * r2806073;
        double r2806076 = 3.0;
        double r2806077 = a;
        double r2806078 = r2806076 * r2806077;
        double r2806079 = c;
        double r2806080 = r2806078 * r2806079;
        double r2806081 = r2806075 - r2806080;
        double r2806082 = sqrt(r2806081);
        double r2806083 = r2806074 + r2806082;
        double r2806084 = r2806083 / r2806078;
        return r2806084;
}
↓
double f(double a, double b, double c) {
        double r2806085 = b;
        double r2806086 = 6.961131476357277e-08;
        bool r2806087 = r2806085 <= r2806086;
        double r2806088 = r2806085 * r2806085;
        double r2806089 = c;
        double r2806090 = a;
        double r2806091 = 3.0;
        double r2806092 = r2806090 * r2806091;
        double r2806093 = r2806089 * r2806092;
        double r2806094 = r2806088 - r2806093;
        double r2806095 = sqrt(r2806094);
        double r2806096 = r2806094 * r2806095;
        double r2806097 = r2806088 * r2806085;
        double r2806098 = r2806096 - r2806097;
        double r2806099 = r2806085 * r2806095;
        double r2806100 = r2806099 + r2806088;
        double r2806101 = r2806094 + r2806100;
        double r2806102 = r2806098 / r2806101;
        double r2806103 = r2806102 / r2806092;
        double r2806104 = -0.5;
        double r2806105 = r2806089 / r2806085;
        double r2806106 = r2806104 * r2806105;
        double r2806107 = r2806087 ? r2806103 : r2806106;
        return r2806107;
}

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 < 6.961131476357277e-08

    1. Initial program 12.8

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

    2. Simplified12.8

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

    4. Applied flip3--13.0

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

    5. Simplified12.5

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

    6. Simplified12.5

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

    if 6.961131476357277e-08 < b

    1. Initial program 44.8

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

    2. Simplified44.8

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

    3. Taylor expanded around inf 11.3

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

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 6.961131476357276728544534868600712762543 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - c \cdot \left(a \cdot 3\right)\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - c \cdot \left(a \cdot 3\right)\right) + \left(b \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} + b \cdot b\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(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) (* (* 3.0 a) c)))) (* 3.0 a)))