Average Error: 43.9 → 11.4
Time: 18.9s
Precision: 64
\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 6.961131476357276728544534868600712762543 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{\sqrt[3]{2}}}{\sqrt{a}} \cdot \frac{\frac{1}{\sqrt[3]{2} \cdot \sqrt[3]{2}}}{\sqrt{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le 6.961131476357276728544534868600712762543 \cdot 10^{-8}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{\sqrt[3]{2}}}{\sqrt{a}} \cdot \frac{\frac{1}{\sqrt[3]{2} \cdot \sqrt[3]{2}}}{\sqrt{a}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1852283 = b;
        double r1852284 = -r1852283;
        double r1852285 = r1852283 * r1852283;
        double r1852286 = 4.0;
        double r1852287 = a;
        double r1852288 = r1852286 * r1852287;
        double r1852289 = c;
        double r1852290 = r1852288 * r1852289;
        double r1852291 = r1852285 - r1852290;
        double r1852292 = sqrt(r1852291);
        double r1852293 = r1852284 + r1852292;
        double r1852294 = 2.0;
        double r1852295 = r1852294 * r1852287;
        double r1852296 = r1852293 / r1852295;
        return r1852296;
}
↓
double f(double a, double b, double c) {
        double r1852297 = b;
        double r1852298 = 6.961131476357277e-08;
        bool r1852299 = r1852297 <= r1852298;
        double r1852300 = r1852297 * r1852297;
        double r1852301 = 4.0;
        double r1852302 = c;
        double r1852303 = a;
        double r1852304 = r1852302 * r1852303;
        double r1852305 = r1852301 * r1852304;
        double r1852306 = r1852300 - r1852305;
        double r1852307 = sqrt(r1852306);
        double r1852308 = r1852307 - r1852297;
        double r1852309 = 2.0;
        double r1852310 = cbrt(r1852309);
        double r1852311 = r1852308 / r1852310;
        double r1852312 = sqrt(r1852303);
        double r1852313 = r1852311 / r1852312;
        double r1852314 = 1.0;
        double r1852315 = r1852310 * r1852310;
        double r1852316 = r1852314 / r1852315;
        double r1852317 = r1852316 / r1852312;
        double r1852318 = r1852313 * r1852317;
        double r1852319 = r1852302 / r1852297;
        double r1852320 = -1.0;
        double r1852321 = r1852319 * r1852320;
        double r1852322 = r1852299 ? r1852318 : r1852321;
        return r1852322;
}

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

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Simplified12.8

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2}}{a}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt12.8

      \[\leadsto \frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2}}{\color{blue}{\sqrt{a} \cdot \sqrt{a}}}\]

    5. Applied add-cube-cbrt12.9

      \[\leadsto \frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{\color{blue}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{2}}}}{\sqrt{a} \cdot \sqrt{a}}\]

    6. Applied *-un-lft-identity12.9

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b\right)}}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{2}}}{\sqrt{a} \cdot \sqrt{a}}\]

    7. Applied times-frac12.9

      \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt[3]{2} \cdot \sqrt[3]{2}} \cdot \frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{\sqrt[3]{2}}}}{\sqrt{a} \cdot \sqrt{a}}\]

    8. Applied times-frac12.9

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt[3]{2} \cdot \sqrt[3]{2}}}{\sqrt{a}} \cdot \frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{\sqrt[3]{2}}}{\sqrt{a}}}\]

    if 6.961131476357277e-08 < b

    1. Initial program 44.7

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Simplified44.7

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2}}{a}}\]

    3. Taylor expanded around inf 11.4

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

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 6.961131476357276728544534868600712762543 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{\sqrt[3]{2}}}{\sqrt{a}} \cdot \frac{\frac{1}{\sqrt[3]{2} \cdot \sqrt[3]{2}}}{\sqrt{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4.0 a) c)))) (* 2.0 a)))