Average Error: 43.4 → 11.5
Time: 35.6s
Precision: 64
\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.00125353222550368486286342939450833000592:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}{\sqrt[3]{a}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}{\sqrt[3]{a}}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1497536 = b;
        double r1497537 = -r1497536;
        double r1497538 = r1497536 * r1497536;
        double r1497539 = 4.0;
        double r1497540 = a;
        double r1497541 = r1497539 * r1497540;
        double r1497542 = c;
        double r1497543 = r1497541 * r1497542;
        double r1497544 = r1497538 - r1497543;
        double r1497545 = sqrt(r1497544);
        double r1497546 = r1497537 + r1497545;
        double r1497547 = 2.0;
        double r1497548 = r1497547 * r1497540;
        double r1497549 = r1497546 / r1497548;
        return r1497549;
}
↓
double f(double a, double b, double c) {
        double r1497550 = b;
        double r1497551 = 0.0012535322255036849;
        bool r1497552 = r1497550 <= r1497551;
        double r1497553 = 1.0;
        double r1497554 = a;
        double r1497555 = cbrt(r1497554);
        double r1497556 = r1497555 * r1497555;
        double r1497557 = r1497553 / r1497556;
        double r1497558 = r1497550 * r1497550;
        double r1497559 = c;
        double r1497560 = r1497554 * r1497559;
        double r1497561 = 4.0;
        double r1497562 = r1497560 * r1497561;
        double r1497563 = r1497558 - r1497562;
        double r1497564 = sqrt(r1497563);
        double r1497565 = r1497564 - r1497550;
        double r1497566 = r1497565 / r1497555;
        double r1497567 = r1497557 * r1497566;
        double r1497568 = 2.0;
        double r1497569 = r1497567 / r1497568;
        double r1497570 = -2.0;
        double r1497571 = r1497559 / r1497550;
        double r1497572 = r1497570 * r1497571;
        double r1497573 = r1497572 / r1497568;
        double r1497574 = r1497552 ? r1497569 : r1497573;
        return r1497574;
}

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

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

    4. Applied add-cube-cbrt20.3

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

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

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

    6. Applied times-frac20.3

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

    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

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

    3. Taylor expanded around inf 10.6

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

  4. Final simplification11.5

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

Reproduce

herbie shell --seed 2019168 
(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)))