Average Error: 34.2 → 10.0
Time: 4.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.2389466313579672 \cdot 10^{127}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.6670468245058271 \cdot 10^{-85}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -5.2389466313579672 \cdot 10^{127}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.6670468245058271 \cdot 10^{-85}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r50721 = b;
        double r50722 = -r50721;
        double r50723 = r50721 * r50721;
        double r50724 = 4.0;
        double r50725 = a;
        double r50726 = r50724 * r50725;
        double r50727 = c;
        double r50728 = r50726 * r50727;
        double r50729 = r50723 - r50728;
        double r50730 = sqrt(r50729);
        double r50731 = r50722 + r50730;
        double r50732 = 2.0;
        double r50733 = r50732 * r50725;
        double r50734 = r50731 / r50733;
        return r50734;
}
↓
double f(double a, double b, double c) {
        double r50735 = b;
        double r50736 = -5.238946631357967e+127;
        bool r50737 = r50735 <= r50736;
        double r50738 = 1.0;
        double r50739 = c;
        double r50740 = r50739 / r50735;
        double r50741 = a;
        double r50742 = r50735 / r50741;
        double r50743 = r50740 - r50742;
        double r50744 = r50738 * r50743;
        double r50745 = 1.667046824505827e-85;
        bool r50746 = r50735 <= r50745;
        double r50747 = r50735 * r50735;
        double r50748 = 4.0;
        double r50749 = r50748 * r50741;
        double r50750 = r50749 * r50739;
        double r50751 = r50747 - r50750;
        double r50752 = sqrt(r50751);
        double r50753 = -r50735;
        double r50754 = r50752 + r50753;
        double r50755 = 2.0;
        double r50756 = r50755 * r50741;
        double r50757 = r50754 / r50756;
        double r50758 = -1.0;
        double r50759 = r50758 * r50740;
        double r50760 = r50746 ? r50757 : r50759;
        double r50761 = r50737 ? r50744 : r50760;
        return r50761;
}

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 < -5.238946631357967e+127

    1. Initial program 54.2

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

    2. Taylor expanded around -inf 3.3

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

    3. Simplified3.3

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

    if -5.238946631357967e+127 < b < 1.667046824505827e-85

    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 +-commutative12.2

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

    if 1.667046824505827e-85 < b

    1. Initial program 52.8

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

    2. Taylor expanded around inf 9.7

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.2389466313579672 \cdot 10^{127}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.6670468245058271 \cdot 10^{-85}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))