Average Error: 34.0 → 9.6
Time: 5.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.5940112039867074 \cdot 10^{100}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.267195199467958 \cdot 10^{-82}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -3.5940112039867074 \cdot 10^{100}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 2.267195199467958 \cdot 10^{-82}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r59789 = b;
        double r59790 = -r59789;
        double r59791 = r59789 * r59789;
        double r59792 = 4.0;
        double r59793 = a;
        double r59794 = r59792 * r59793;
        double r59795 = c;
        double r59796 = r59794 * r59795;
        double r59797 = r59791 - r59796;
        double r59798 = sqrt(r59797);
        double r59799 = r59790 + r59798;
        double r59800 = 2.0;
        double r59801 = r59800 * r59793;
        double r59802 = r59799 / r59801;
        return r59802;
}
↓
double f(double a, double b, double c) {
        double r59803 = b;
        double r59804 = -3.5940112039867074e+100;
        bool r59805 = r59803 <= r59804;
        double r59806 = 1.0;
        double r59807 = c;
        double r59808 = r59807 / r59803;
        double r59809 = a;
        double r59810 = r59803 / r59809;
        double r59811 = r59808 - r59810;
        double r59812 = r59806 * r59811;
        double r59813 = 2.267195199467958e-82;
        bool r59814 = r59803 <= r59813;
        double r59815 = 1.0;
        double r59816 = 2.0;
        double r59817 = r59816 * r59809;
        double r59818 = -r59803;
        double r59819 = r59803 * r59803;
        double r59820 = 4.0;
        double r59821 = r59820 * r59809;
        double r59822 = r59821 * r59807;
        double r59823 = r59819 - r59822;
        double r59824 = sqrt(r59823);
        double r59825 = r59818 + r59824;
        double r59826 = r59817 / r59825;
        double r59827 = r59815 / r59826;
        double r59828 = -1.0;
        double r59829 = r59828 * r59808;
        double r59830 = r59814 ? r59827 : r59829;
        double r59831 = r59805 ? r59812 : r59830;
        return r59831;
}

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 < -3.5940112039867074e+100

    1. Initial program 47.3

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

    2. Taylor expanded around -inf 3.8

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

    3. Simplified3.8

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

    if -3.5940112039867074e+100 < b < 2.267195199467958e-82

    1. Initial program 12.0

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

    3. Applied clear-num12.1

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

    if 2.267195199467958e-82 < b

    1. Initial program 52.9

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

    2. Taylor expanded around inf 9.0

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

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.5940112039867074 \cdot 10^{100}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.267195199467958 \cdot 10^{-82}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))