Average Error: 34.4 → 10.6
Time: 6.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.9644058459680186 \cdot 10^{71}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.05029242402897421 \cdot 10^{-108}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.9644058459680186 \cdot 10^{71}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.05029242402897421 \cdot 10^{-108}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r46955 = b;
        double r46956 = -r46955;
        double r46957 = r46955 * r46955;
        double r46958 = 4.0;
        double r46959 = a;
        double r46960 = c;
        double r46961 = r46959 * r46960;
        double r46962 = r46958 * r46961;
        double r46963 = r46957 - r46962;
        double r46964 = sqrt(r46963);
        double r46965 = r46956 + r46964;
        double r46966 = 2.0;
        double r46967 = r46966 * r46959;
        double r46968 = r46965 / r46967;
        return r46968;
}
↓
double f(double a, double b, double c) {
        double r46969 = b;
        double r46970 = -2.9644058459680186e+71;
        bool r46971 = r46969 <= r46970;
        double r46972 = 1.0;
        double r46973 = c;
        double r46974 = r46973 / r46969;
        double r46975 = a;
        double r46976 = r46969 / r46975;
        double r46977 = r46974 - r46976;
        double r46978 = r46972 * r46977;
        double r46979 = 1.0502924240289742e-108;
        bool r46980 = r46969 <= r46979;
        double r46981 = -r46969;
        double r46982 = r46969 * r46969;
        double r46983 = 4.0;
        double r46984 = r46975 * r46973;
        double r46985 = r46983 * r46984;
        double r46986 = r46982 - r46985;
        double r46987 = sqrt(r46986);
        double r46988 = r46981 + r46987;
        double r46989 = 1.0;
        double r46990 = 2.0;
        double r46991 = r46990 * r46975;
        double r46992 = r46989 / r46991;
        double r46993 = r46988 * r46992;
        double r46994 = -1.0;
        double r46995 = r46994 * r46974;
        double r46996 = r46980 ? r46993 : r46995;
        double r46997 = r46971 ? r46978 : r46996;
        return r46997;
}

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

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

Derivation

  1. Split input into 3 regimes

  2. if b < -2.9644058459680186e+71

    1. Initial program 42.3

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

    2. Taylor expanded around -inf 4.7

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

    3. Simplified4.7

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

    if -2.9644058459680186e+71 < b < 1.0502924240289742e-108

    1. Initial program 13.1

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

    3. Applied div-inv13.2

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

    if 1.0502924240289742e-108 < b

    1. Initial program 51.6

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

    2. Taylor expanded around inf 10.7

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.9644058459680186 \cdot 10^{71}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.05029242402897421 \cdot 10^{-108}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(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)))