Average Error: 33.8 → 10.3
Time: 18.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.025649824816678368861606895534923213042 \cdot 10^{153}:\\ \;\;\;\;\left(\frac{c \cdot 1}{b} - 0.5 \cdot \frac{b}{a}\right) - \frac{\frac{b}{2}}{a}\\ \mathbf{elif}\;b \le 3.047677256636077515553757160900796353717 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a} - \frac{\frac{b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]
\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 -2.025649824816678368861606895534923213042 \cdot 10^{153}:\\
\;\;\;\;\left(\frac{c \cdot 1}{b} - 0.5 \cdot \frac{b}{a}\right) - \frac{\frac{b}{2}}{a}\\

\mathbf{elif}\;b \le 3.047677256636077515553757160900796353717 \cdot 10^{-81}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a} - \frac{\frac{b}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r70876 = b;
        double r70877 = -r70876;
        double r70878 = r70876 * r70876;
        double r70879 = 4.0;
        double r70880 = a;
        double r70881 = c;
        double r70882 = r70880 * r70881;
        double r70883 = r70879 * r70882;
        double r70884 = r70878 - r70883;
        double r70885 = sqrt(r70884);
        double r70886 = r70877 + r70885;
        double r70887 = 2.0;
        double r70888 = r70887 * r70880;
        double r70889 = r70886 / r70888;
        return r70889;
}
↓
double f(double a, double b, double c) {
        double r70890 = b;
        double r70891 = -2.0256498248166784e+153;
        bool r70892 = r70890 <= r70891;
        double r70893 = c;
        double r70894 = 1.0;
        double r70895 = r70893 * r70894;
        double r70896 = r70895 / r70890;
        double r70897 = 0.5;
        double r70898 = a;
        double r70899 = r70890 / r70898;
        double r70900 = r70897 * r70899;
        double r70901 = r70896 - r70900;
        double r70902 = 2.0;
        double r70903 = r70890 / r70902;
        double r70904 = r70903 / r70898;
        double r70905 = r70901 - r70904;
        double r70906 = 3.0476772566360775e-81;
        bool r70907 = r70890 <= r70906;
        double r70908 = r70890 * r70890;
        double r70909 = 4.0;
        double r70910 = r70893 * r70898;
        double r70911 = r70909 * r70910;
        double r70912 = r70908 - r70911;
        double r70913 = sqrt(r70912);
        double r70914 = r70902 * r70898;
        double r70915 = r70913 / r70914;
        double r70916 = r70915 - r70904;
        double r70917 = r70893 / r70890;
        double r70918 = -1.0;
        double r70919 = r70917 * r70918;
        double r70920 = r70907 ? r70916 : r70919;
        double r70921 = r70892 ? r70905 : r70920;
        return r70921;
}

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
Target20.6
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.0256498248166784e+153

    1. Initial program 63.6

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

    2. Simplified63.6

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

    4. Applied div-sub63.6

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

    5. Applied div-sub63.6

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

    6. Simplified63.6

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

    7. Taylor expanded around -inf 2.0

      \[\leadsto \color{blue}{\left(1 \cdot \frac{c}{b} - 0.5 \cdot \frac{b}{a}\right)} - \frac{\frac{b}{2}}{a}\]

    8. Simplified2.0

      \[\leadsto \color{blue}{\left(\frac{c \cdot 1}{b} - \frac{b}{a} \cdot 0.5\right)} - \frac{\frac{b}{2}}{a}\]

    if -2.0256498248166784e+153 < b < 3.0476772566360775e-81

    1. Initial program 11.9

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

    2. Simplified11.9

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

    4. Applied div-sub11.9

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

    5. Applied div-sub11.9

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

    6. Simplified11.9

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

    if 3.0476772566360775e-81 < b

    1. Initial program 52.2

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

    2. Simplified52.2

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

    4. Applied div-sub52.2

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

    5. Applied div-sub52.9

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

    6. Simplified52.9

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

    7. Taylor expanded around inf 10.5

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

    8. Simplified10.5

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.025649824816678368861606895534923213042 \cdot 10^{153}:\\ \;\;\;\;\left(\frac{c \cdot 1}{b} - 0.5 \cdot \frac{b}{a}\right) - \frac{\frac{b}{2}}{a}\\ \mathbf{elif}\;b \le 3.047677256636077515553757160900796353717 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a} - \frac{\frac{b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(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)))