Average Error: 43.4 → 11.5
Time: 44.0s
Precision: 64
\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.00125353222550368486286342939450833000592:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{\sqrt{2}}}{\sqrt[3]{a}} \cdot \frac{\frac{1}{\sqrt{2}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]
\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 0.00125353222550368486286342939450833000592:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{\sqrt{2}}}{\sqrt[3]{a}} \cdot \frac{\frac{1}{\sqrt{2}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1911516 = b;
        double r1911517 = -r1911516;
        double r1911518 = r1911516 * r1911516;
        double r1911519 = 4.0;
        double r1911520 = a;
        double r1911521 = r1911519 * r1911520;
        double r1911522 = c;
        double r1911523 = r1911521 * r1911522;
        double r1911524 = r1911518 - r1911523;
        double r1911525 = sqrt(r1911524);
        double r1911526 = r1911517 + r1911525;
        double r1911527 = 2.0;
        double r1911528 = r1911527 * r1911520;
        double r1911529 = r1911526 / r1911528;
        return r1911529;
}
↓
double f(double a, double b, double c) {
        double r1911530 = b;
        double r1911531 = 0.0012535322255036849;
        bool r1911532 = r1911530 <= r1911531;
        double r1911533 = r1911530 * r1911530;
        double r1911534 = 4.0;
        double r1911535 = a;
        double r1911536 = c;
        double r1911537 = r1911535 * r1911536;
        double r1911538 = r1911534 * r1911537;
        double r1911539 = r1911533 - r1911538;
        double r1911540 = sqrt(r1911539);
        double r1911541 = r1911540 - r1911530;
        double r1911542 = 2.0;
        double r1911543 = sqrt(r1911542);
        double r1911544 = r1911541 / r1911543;
        double r1911545 = cbrt(r1911535);
        double r1911546 = r1911544 / r1911545;
        double r1911547 = 1.0;
        double r1911548 = r1911547 / r1911543;
        double r1911549 = r1911545 * r1911545;
        double r1911550 = r1911548 / r1911549;
        double r1911551 = r1911546 * r1911550;
        double r1911552 = r1911536 / r1911530;
        double r1911553 = -1.0;
        double r1911554 = r1911552 * r1911553;
        double r1911555 = r1911532 ? r1911551 : r1911554;
        return r1911555;
}

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 < 0.0012535322255036849

    1. Initial program 20.2

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

    2. Simplified20.2

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

    4. Applied add-cube-cbrt20.3

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

    5. Applied add-sqr-sqrt20.3

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

    6. Applied *-un-lft-identity20.3

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

    7. Applied times-frac20.2

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

    8. Applied times-frac20.3

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

    if 0.0012535322255036849 < b

    1. Initial program 45.7

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

    2. Simplified45.7

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

    3. Taylor expanded around inf 10.6

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

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.00125353222550368486286342939450833000592:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{\sqrt{2}}}{\sqrt[3]{a}} \cdot \frac{\frac{1}{\sqrt{2}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(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)))