Average Error: 34.2 → 10.0
Time: 5.5s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.01157973271056712 \cdot 10^{-81}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.3176462918432122 \cdot 10^{99}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -4.01157973271056712 \cdot 10^{-81}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.3176462918432122 \cdot 10^{99}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r75825 = b;
        double r75826 = -r75825;
        double r75827 = r75825 * r75825;
        double r75828 = 4.0;
        double r75829 = a;
        double r75830 = c;
        double r75831 = r75829 * r75830;
        double r75832 = r75828 * r75831;
        double r75833 = r75827 - r75832;
        double r75834 = sqrt(r75833);
        double r75835 = r75826 - r75834;
        double r75836 = 2.0;
        double r75837 = r75836 * r75829;
        double r75838 = r75835 / r75837;
        return r75838;
}
↓
double f(double a, double b, double c) {
        double r75839 = b;
        double r75840 = -4.011579732710567e-81;
        bool r75841 = r75839 <= r75840;
        double r75842 = -1.0;
        double r75843 = c;
        double r75844 = r75843 / r75839;
        double r75845 = r75842 * r75844;
        double r75846 = 1.3176462918432122e+99;
        bool r75847 = r75839 <= r75846;
        double r75848 = -r75839;
        double r75849 = 2.0;
        double r75850 = a;
        double r75851 = r75849 * r75850;
        double r75852 = r75848 / r75851;
        double r75853 = r75839 * r75839;
        double r75854 = 4.0;
        double r75855 = r75850 * r75843;
        double r75856 = r75854 * r75855;
        double r75857 = r75853 - r75856;
        double r75858 = sqrt(r75857);
        double r75859 = r75858 / r75851;
        double r75860 = r75852 - r75859;
        double r75861 = 1.0;
        double r75862 = r75839 / r75850;
        double r75863 = r75844 - r75862;
        double r75864 = r75861 * r75863;
        double r75865 = r75847 ? r75860 : r75864;
        double r75866 = r75841 ? r75845 : r75865;
        return r75866;
}

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

Target

Original34.2
Target21.5
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.011579732710567e-81

    1. Initial program 52.8

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

    2. Taylor expanded around -inf 9.4

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

    if -4.011579732710567e-81 < b < 1.3176462918432122e+99

    1. Initial program 12.9

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

    3. Applied div-sub12.9

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

    if 1.3176462918432122e+99 < b

    1. Initial program 46.8

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

    2. Taylor expanded around inf 3.8

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

    3. Simplified3.8

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.01157973271056712 \cdot 10^{-81}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.3176462918432122 \cdot 10^{99}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))