Average Error: 34.3 → 10.3
Time: 8.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.525509213201143173482122987435994097044 \cdot 10^{-76}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.112435874232008033470805388544595401902 \cdot 10^{74}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -1.525509213201143173482122987435994097044 \cdot 10^{-76}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.112435874232008033470805388544595401902 \cdot 10^{74}:\\
\;\;\;\;\frac{\left(-b\right) - \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 r54853 = b;
        double r54854 = -r54853;
        double r54855 = r54853 * r54853;
        double r54856 = 4.0;
        double r54857 = a;
        double r54858 = c;
        double r54859 = r54857 * r54858;
        double r54860 = r54856 * r54859;
        double r54861 = r54855 - r54860;
        double r54862 = sqrt(r54861);
        double r54863 = r54854 - r54862;
        double r54864 = 2.0;
        double r54865 = r54864 * r54857;
        double r54866 = r54863 / r54865;
        return r54866;
}
↓
double f(double a, double b, double c) {
        double r54867 = b;
        double r54868 = -1.5255092132011432e-76;
        bool r54869 = r54867 <= r54868;
        double r54870 = -1.0;
        double r54871 = c;
        double r54872 = r54871 / r54867;
        double r54873 = r54870 * r54872;
        double r54874 = 1.112435874232008e+74;
        bool r54875 = r54867 <= r54874;
        double r54876 = -r54867;
        double r54877 = r54867 * r54867;
        double r54878 = 4.0;
        double r54879 = a;
        double r54880 = r54879 * r54871;
        double r54881 = r54878 * r54880;
        double r54882 = r54877 - r54881;
        double r54883 = sqrt(r54882);
        double r54884 = r54876 - r54883;
        double r54885 = 2.0;
        double r54886 = r54885 * r54879;
        double r54887 = r54884 / r54886;
        double r54888 = 1.0;
        double r54889 = r54867 / r54879;
        double r54890 = r54872 - r54889;
        double r54891 = r54888 * r54890;
        double r54892 = r54875 ? r54887 : r54891;
        double r54893 = r54869 ? r54873 : r54892;
        return r54893;
}

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

Original34.3
Target21.3
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.5255092132011432e-76

    1. Initial program 53.0

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

    2. Taylor expanded around -inf 9.2

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

    if -1.5255092132011432e-76 < b < 1.112435874232008e+74

    1. Initial program 13.5

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

    if 1.112435874232008e+74 < b

    1. Initial program 43.0

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

    2. Taylor expanded around inf 5.0

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

    3. Simplified5.0

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.525509213201143173482122987435994097044 \cdot 10^{-76}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.112435874232008033470805388544595401902 \cdot 10^{74}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019308 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :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)))