Average Error: 33.8 → 6.7
Time: 9.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.301687926884188663878043402578250574713 \cdot 10^{98}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le \frac{-1077853067741081}{1.365609355853794155331553646739713596855 \cdot 10^{244}}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.208057797080499790536355473922963434123 \cdot 10^{104}:\\ \;\;\;\;\frac{\frac{c}{\frac{2}{4}}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 \frac{-1077853067741081}{1.365609355853794155331553646739713596855 \cdot 10^{244}}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

\mathbf{elif}\;b \le 6.208057797080499790536355473922963434123 \cdot 10^{104}:\\
\;\;\;\;\frac{\frac{c}{\frac{2}{4}}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r58798 = b;
        double r58799 = -r58798;
        double r58800 = r58798 * r58798;
        double r58801 = 4.0;
        double r58802 = a;
        double r58803 = c;
        double r58804 = r58802 * r58803;
        double r58805 = r58801 * r58804;
        double r58806 = r58800 - r58805;
        double r58807 = sqrt(r58806);
        double r58808 = r58799 + r58807;
        double r58809 = 2.0;
        double r58810 = r58809 * r58802;
        double r58811 = r58808 / r58810;
        return r58811;
}
↓
double f(double a, double b, double c) {
        double r58812 = b;
        double r58813 = -8.301687926884189e+98;
        bool r58814 = r58812 <= r58813;
        double r58815 = 1.0;
        double r58816 = c;
        double r58817 = r58816 / r58812;
        double r58818 = a;
        double r58819 = r58812 / r58818;
        double r58820 = r58817 - r58819;
        double r58821 = r58815 * r58820;
        double r58822 = -1077853067741081.0;
        double r58823 = 1.3656093558537942e+244;
        double r58824 = r58822 / r58823;
        bool r58825 = r58812 <= r58824;
        double r58826 = -r58812;
        double r58827 = r58812 * r58812;
        double r58828 = 4.0;
        double r58829 = r58818 * r58816;
        double r58830 = r58828 * r58829;
        double r58831 = r58827 - r58830;
        double r58832 = sqrt(r58831);
        double r58833 = r58826 + r58832;
        double r58834 = 2.0;
        double r58835 = r58834 * r58818;
        double r58836 = r58833 / r58835;
        double r58837 = 6.2080577970805e+104;
        bool r58838 = r58812 <= r58837;
        double r58839 = r58834 / r58828;
        double r58840 = r58816 / r58839;
        double r58841 = r58826 - r58832;
        double r58842 = r58840 / r58841;
        double r58843 = -1.0;
        double r58844 = r58843 * r58817;
        double r58845 = r58838 ? r58842 : r58844;
        double r58846 = r58825 ? r58836 : r58845;
        double r58847 = r58814 ? r58821 : r58846;
        return r58847;
}

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
Herbie6.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 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. Taylor expanded around -inf 3.6

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

    3. Simplified3.6

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

    if -8.301687926884189e+98 < b < -7.892835993842436e-230

    1. Initial program 8.1

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

    if -7.892835993842436e-230 < b < 6.2080577970805e+104

    1. Initial program 29.9

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

    3. Applied flip-+30.0

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

    4. Simplified15.8

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied clear-num15.9

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}}\]

    7. Simplified15.2

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

    9. Applied times-frac15.2

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

    10. Simplified9.8

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

    12. Applied associate-/r*9.5

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

    13. Simplified9.4

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

    if 6.2080577970805e+104 < b

    1. Initial program 59.4

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

    2. Taylor expanded around inf 2.8

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.301687926884188663878043402578250574713 \cdot 10^{98}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le \frac{-1077853067741081}{1.365609355853794155331553646739713596855 \cdot 10^{244}}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.208057797080499790536355473922963434123 \cdot 10^{104}:\\ \;\;\;\;\frac{\frac{c}{\frac{2}{4}}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(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)))