Average Error: 34.2 → 9.9
Time: 6.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.4052299045085703 \cdot 10^{151}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.2529910715609764 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \left(-b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.4052299045085703 \cdot 10^{151}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.2529910715609764 \cdot 10^{-90}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \left(-b\right)}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r82876 = b;
        double r82877 = -r82876;
        double r82878 = r82876 * r82876;
        double r82879 = 4.0;
        double r82880 = a;
        double r82881 = c;
        double r82882 = r82880 * r82881;
        double r82883 = r82879 * r82882;
        double r82884 = r82878 - r82883;
        double r82885 = sqrt(r82884);
        double r82886 = r82877 + r82885;
        double r82887 = 2.0;
        double r82888 = r82887 * r82880;
        double r82889 = r82886 / r82888;
        return r82889;
}
↓
double f(double a, double b, double c) {
        double r82890 = b;
        double r82891 = -1.4052299045085703e+151;
        bool r82892 = r82890 <= r82891;
        double r82893 = 1.0;
        double r82894 = c;
        double r82895 = r82894 / r82890;
        double r82896 = a;
        double r82897 = r82890 / r82896;
        double r82898 = r82895 - r82897;
        double r82899 = r82893 * r82898;
        double r82900 = 1.2529910715609764e-90;
        bool r82901 = r82890 <= r82900;
        double r82902 = r82890 * r82890;
        double r82903 = 4.0;
        double r82904 = r82896 * r82894;
        double r82905 = r82903 * r82904;
        double r82906 = r82902 - r82905;
        double r82907 = sqrt(r82906);
        double r82908 = -r82890;
        double r82909 = r82907 + r82908;
        double r82910 = 2.0;
        double r82911 = r82910 * r82896;
        double r82912 = r82909 / r82911;
        double r82913 = -1.0;
        double r82914 = r82913 * r82895;
        double r82915 = r82901 ? r82912 : r82914;
        double r82916 = r82892 ? r82899 : r82915;
        return r82916;
}

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
Target21.1
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.4052299045085703e+151

    1. Initial program 63.1

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

    2. Taylor expanded around -inf 2.2

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

    3. Simplified2.2

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

    if -1.4052299045085703e+151 < b < 1.2529910715609764e-90

    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 +-commutative12.1

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

    if 1.2529910715609764e-90 < b

    1. Initial program 52.5

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

    2. Taylor expanded around inf 9.5

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.4052299045085703 \cdot 10^{151}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.2529910715609764 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \left(-b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

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

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