Average Error: 33.8 → 10.0
Time: 12.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.301687926884188663878043402578250574713 \cdot 10^{98}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 7.029337360841496098479843453825374035485 \cdot 10^{-56}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -8.301687926884188663878043402578250574713 \cdot 10^{98}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 7.029337360841496098479843453825374035485 \cdot 10^{-56}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r36902 = b;
        double r36903 = -r36902;
        double r36904 = r36902 * r36902;
        double r36905 = 4.0;
        double r36906 = a;
        double r36907 = c;
        double r36908 = r36906 * r36907;
        double r36909 = r36905 * r36908;
        double r36910 = r36904 - r36909;
        double r36911 = sqrt(r36910);
        double r36912 = r36903 + r36911;
        double r36913 = 2.0;
        double r36914 = r36913 * r36906;
        double r36915 = r36912 / r36914;
        return r36915;
}
↓
double f(double a, double b, double c) {
        double r36916 = b;
        double r36917 = -8.301687926884189e+98;
        bool r36918 = r36916 <= r36917;
        double r36919 = 1.0;
        double r36920 = c;
        double r36921 = r36920 / r36916;
        double r36922 = a;
        double r36923 = r36916 / r36922;
        double r36924 = r36921 - r36923;
        double r36925 = r36919 * r36924;
        double r36926 = 7.029337360841496e-56;
        bool r36927 = r36916 <= r36926;
        double r36928 = r36916 * r36916;
        double r36929 = 4.0;
        double r36930 = r36922 * r36920;
        double r36931 = r36929 * r36930;
        double r36932 = r36928 - r36931;
        double r36933 = sqrt(r36932);
        double r36934 = r36933 - r36916;
        double r36935 = 2.0;
        double r36936 = r36935 * r36922;
        double r36937 = r36934 / r36936;
        double r36938 = -1.0;
        double r36939 = r36938 * r36921;
        double r36940 = r36927 ? r36937 : r36939;
        double r36941 = r36918 ? r36925 : r36940;
        return r36941;
}

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

Original33.8
Target20.8
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -8.301687926884189e+98

    1. Initial program 46.2

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

    2. Simplified46.2

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

    3. Taylor expanded around -inf 3.6

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

    4. Simplified3.6

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

    if -8.301687926884189e+98 < b < 7.029337360841496e-56

    1. Initial program 13.5

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

    2. Simplified13.5

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

    4. Applied *-un-lft-identity13.5

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

    if 7.029337360841496e-56 < b

    1. Initial program 53.8

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

    2. Simplified53.8

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

    3. Taylor expanded around inf 8.2

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.301687926884188663878043402578250574713 \cdot 10^{98}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 7.029337360841496098479843453825374035485 \cdot 10^{-56}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +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)))