Average Error: 34.9 → 10.1
Time: 19.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.680329042988888396603264581948851078331 \cdot 10^{148}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 4.612990823111230552052602417245542305295 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{2} \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]
\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 -3.680329042988888396603264581948851078331 \cdot 10^{148}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\mathbf{elif}\;b \le 4.612990823111230552052602417245542305295 \cdot 10^{-104}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{2} \cdot \frac{1}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2046620 = b;
        double r2046621 = -r2046620;
        double r2046622 = r2046620 * r2046620;
        double r2046623 = 4.0;
        double r2046624 = a;
        double r2046625 = r2046623 * r2046624;
        double r2046626 = c;
        double r2046627 = r2046625 * r2046626;
        double r2046628 = r2046622 - r2046627;
        double r2046629 = sqrt(r2046628);
        double r2046630 = r2046621 + r2046629;
        double r2046631 = 2.0;
        double r2046632 = r2046631 * r2046624;
        double r2046633 = r2046630 / r2046632;
        return r2046633;
}
↓
double f(double a, double b, double c) {
        double r2046634 = b;
        double r2046635 = -3.6803290429888884e+148;
        bool r2046636 = r2046634 <= r2046635;
        double r2046637 = c;
        double r2046638 = r2046637 / r2046634;
        double r2046639 = a;
        double r2046640 = r2046634 / r2046639;
        double r2046641 = r2046638 - r2046640;
        double r2046642 = 1.0;
        double r2046643 = r2046641 * r2046642;
        double r2046644 = 4.6129908231112306e-104;
        bool r2046645 = r2046634 <= r2046644;
        double r2046646 = r2046634 * r2046634;
        double r2046647 = 4.0;
        double r2046648 = r2046637 * r2046639;
        double r2046649 = r2046647 * r2046648;
        double r2046650 = r2046646 - r2046649;
        double r2046651 = sqrt(r2046650);
        double r2046652 = r2046651 - r2046634;
        double r2046653 = 2.0;
        double r2046654 = r2046652 / r2046653;
        double r2046655 = 1.0;
        double r2046656 = r2046655 / r2046639;
        double r2046657 = r2046654 * r2046656;
        double r2046658 = -1.0;
        double r2046659 = r2046638 * r2046658;
        double r2046660 = r2046645 ? r2046657 : r2046659;
        double r2046661 = r2046636 ? r2046643 : r2046660;
        return r2046661;
}

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 3 regimes

  2. if b < -3.6803290429888884e+148

    1. Initial program 62.1

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

    2. Taylor expanded around -inf 2.3

      \[\leadsto \color{blue}{1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}}\]

    3. Simplified2.3

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

    if -3.6803290429888884e+148 < b < 4.6129908231112306e-104

    1. Initial program 12.2

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

    3. Applied clear-num12.3

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

    4. Simplified12.3

      \[\leadsto \frac{1}{\color{blue}{\frac{a}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2}}}}\]
    5. Using strategy rm

    6. Applied div-inv12.3

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

    7. Applied add-cube-cbrt12.3

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

    8. Applied times-frac12.3

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

    9. Simplified12.3

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

    10. Simplified12.3

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

    if 4.6129908231112306e-104 < b

    1. Initial program 52.7

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

    2. Taylor expanded around inf 9.8

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.680329042988888396603264581948851078331 \cdot 10^{148}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 4.612990823111230552052602417245542305295 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{2} \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))