Average Error: 34.0 → 10.6
Time: 4.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.731633690849517820308375807349583220341 \cdot 10^{-121}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.02738286211209785784187544728837722875 \cdot 10^{63}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -2.731633690849517820308375807349583220341 \cdot 10^{-121}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.02738286211209785784187544728837722875 \cdot 10^{63}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r72837 = b;
        double r72838 = -r72837;
        double r72839 = r72837 * r72837;
        double r72840 = 4.0;
        double r72841 = a;
        double r72842 = c;
        double r72843 = r72841 * r72842;
        double r72844 = r72840 * r72843;
        double r72845 = r72839 - r72844;
        double r72846 = sqrt(r72845);
        double r72847 = r72838 - r72846;
        double r72848 = 2.0;
        double r72849 = r72848 * r72841;
        double r72850 = r72847 / r72849;
        return r72850;
}
↓
double f(double a, double b, double c) {
        double r72851 = b;
        double r72852 = -2.731633690849518e-121;
        bool r72853 = r72851 <= r72852;
        double r72854 = -1.0;
        double r72855 = c;
        double r72856 = r72855 / r72851;
        double r72857 = r72854 * r72856;
        double r72858 = 1.0273828621120979e+63;
        bool r72859 = r72851 <= r72858;
        double r72860 = 1.0;
        double r72861 = 2.0;
        double r72862 = a;
        double r72863 = r72861 * r72862;
        double r72864 = -r72851;
        double r72865 = r72851 * r72851;
        double r72866 = 4.0;
        double r72867 = r72862 * r72855;
        double r72868 = r72866 * r72867;
        double r72869 = r72865 - r72868;
        double r72870 = sqrt(r72869);
        double r72871 = r72864 - r72870;
        double r72872 = r72863 / r72871;
        double r72873 = r72860 / r72872;
        double r72874 = 1.0;
        double r72875 = r72851 / r72862;
        double r72876 = r72856 - r72875;
        double r72877 = r72874 * r72876;
        double r72878 = r72859 ? r72873 : r72877;
        double r72879 = r72853 ? r72857 : r72878;
        return r72879;
}

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

Original34.0
Target21.0
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.731633690849518e-121

    1. Initial program 51.0

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

    2. Taylor expanded around -inf 11.5

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

    if -2.731633690849518e-121 < b < 1.0273828621120979e+63

    1. Initial program 12.1

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

    3. Applied clear-num12.2

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

    if 1.0273828621120979e+63 < b

    1. Initial program 39.8

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

    2. Taylor expanded around inf 5.4

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

    3. Simplified5.4

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.731633690849517820308375807349583220341 \cdot 10^{-121}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.02738286211209785784187544728837722875 \cdot 10^{63}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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)))