Average Error: 32.9 → 10.3
Time: 16.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.088000531423294 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 9.354082991670835 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a}}{2} - \frac{\frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -9.088000531423294 \cdot 10^{+152}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 9.354082991670835 \cdot 10^{-125}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a}}{2} - \frac{\frac{b}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r663726 = b;
        double r663727 = -r663726;
        double r663728 = r663726 * r663726;
        double r663729 = 4.0;
        double r663730 = a;
        double r663731 = r663729 * r663730;
        double r663732 = c;
        double r663733 = r663731 * r663732;
        double r663734 = r663728 - r663733;
        double r663735 = sqrt(r663734);
        double r663736 = r663727 + r663735;
        double r663737 = 2.0;
        double r663738 = r663737 * r663730;
        double r663739 = r663736 / r663738;
        return r663739;
}
↓
double f(double a, double b, double c) {
        double r663740 = b;
        double r663741 = -9.088000531423294e+152;
        bool r663742 = r663740 <= r663741;
        double r663743 = c;
        double r663744 = r663743 / r663740;
        double r663745 = a;
        double r663746 = r663740 / r663745;
        double r663747 = r663744 - r663746;
        double r663748 = 9.354082991670835e-125;
        bool r663749 = r663740 <= r663748;
        double r663750 = r663740 * r663740;
        double r663751 = r663743 * r663745;
        double r663752 = 4.0;
        double r663753 = r663751 * r663752;
        double r663754 = r663750 - r663753;
        double r663755 = sqrt(r663754);
        double r663756 = r663755 / r663745;
        double r663757 = 2.0;
        double r663758 = r663756 / r663757;
        double r663759 = r663746 / r663757;
        double r663760 = r663758 - r663759;
        double r663761 = -r663744;
        double r663762 = r663749 ? r663760 : r663761;
        double r663763 = r663742 ? r663747 : r663762;
        return r663763;
}

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 < -9.088000531423294e+152

    1. Initial program 60.4

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

    2. Simplified60.4

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

    3. Taylor expanded around -inf 1.5

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

    if -9.088000531423294e+152 < b < 9.354082991670835e-125

    1. Initial program 10.9

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

    2. Simplified10.9

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

    4. Applied div-sub10.9

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

    5. Applied div-sub10.9

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

    if 9.354082991670835e-125 < b

    1. Initial program 49.8

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

    2. Simplified49.8

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

    3. Taylor expanded around inf 11.9

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

    4. Simplified11.9

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.088000531423294 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 9.354082991670835 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a}}{2} - \frac{\frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))