Average Error: 34.6 → 10.5
Time: 20.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.6581383089037873 \cdot 10^{81}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.45811587950602871 \cdot 10^{-136}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}} - \frac{b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.6581383089037873 \cdot 10^{81}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 2.45811587950602871 \cdot 10^{-136}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}} - \frac{b}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r81917 = b;
        double r81918 = -r81917;
        double r81919 = r81917 * r81917;
        double r81920 = 4.0;
        double r81921 = a;
        double r81922 = c;
        double r81923 = r81921 * r81922;
        double r81924 = r81920 * r81923;
        double r81925 = r81919 - r81924;
        double r81926 = sqrt(r81925);
        double r81927 = r81918 + r81926;
        double r81928 = 2.0;
        double r81929 = r81928 * r81921;
        double r81930 = r81927 / r81929;
        return r81930;
}
↓
double f(double a, double b, double c) {
        double r81931 = b;
        double r81932 = -1.6581383089037873e+81;
        bool r81933 = r81931 <= r81932;
        double r81934 = 1.0;
        double r81935 = c;
        double r81936 = r81935 / r81931;
        double r81937 = a;
        double r81938 = r81931 / r81937;
        double r81939 = r81936 - r81938;
        double r81940 = r81934 * r81939;
        double r81941 = 2.4581158795060287e-136;
        bool r81942 = r81931 <= r81941;
        double r81943 = 1.0;
        double r81944 = 2.0;
        double r81945 = r81944 * r81937;
        double r81946 = r81931 * r81931;
        double r81947 = 4.0;
        double r81948 = r81937 * r81935;
        double r81949 = r81947 * r81948;
        double r81950 = r81946 - r81949;
        double r81951 = sqrt(r81950);
        double r81952 = r81945 / r81951;
        double r81953 = r81943 / r81952;
        double r81954 = r81931 / r81945;
        double r81955 = r81953 - r81954;
        double r81956 = -1.0;
        double r81957 = r81956 * r81936;
        double r81958 = r81942 ? r81955 : r81957;
        double r81959 = r81933 ? r81940 : r81958;
        return r81959;
}

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

Original34.6
Target21.3
Herbie10.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.6581383089037873e+81

    1. Initial program 43.9

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

    2. Simplified43.9

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

    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 -1.6581383089037873e+81 < b < 2.4581158795060287e-136

    1. Initial program 11.7

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

    2. Simplified11.7

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

    4. Applied div-sub11.7

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

    6. Applied *-un-lft-identity11.7

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

    7. Applied sqrt-prod11.7

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

    8. Applied associate-/l*11.8

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

    if 2.4581158795060287e-136 < b

    1. Initial program 50.8

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

    2. Simplified50.8

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

    3. Taylor expanded around inf 12.0

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.6581383089037873 \cdot 10^{81}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.45811587950602871 \cdot 10^{-136}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}} - \frac{b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))