Average Error: 33.7 → 9.0
Time: 22.9s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6239046376.848015:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -2.396838732613953 \cdot 10^{-306}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{a} \cdot \left(\left(c \cdot 4\right) \cdot a\right)}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\\ \mathbf{elif}\;b \le 9.179168538250646 \cdot 10^{+63}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]
\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 -6239046376.848015:\\
\;\;\;\;\frac{-c}{b}\\

\mathbf{elif}\;b \le -2.396838732613953 \cdot 10^{-306}:\\
\;\;\;\;\frac{\frac{\frac{1}{2}}{a} \cdot \left(\left(c \cdot 4\right) \cdot a\right)}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\\

\mathbf{elif}\;b \le 9.179168538250646 \cdot 10^{+63}:\\
\;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{-b}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r3421775 = b;
        double r3421776 = -r3421775;
        double r3421777 = r3421775 * r3421775;
        double r3421778 = 4.0;
        double r3421779 = a;
        double r3421780 = c;
        double r3421781 = r3421779 * r3421780;
        double r3421782 = r3421778 * r3421781;
        double r3421783 = r3421777 - r3421782;
        double r3421784 = sqrt(r3421783);
        double r3421785 = r3421776 - r3421784;
        double r3421786 = 2.0;
        double r3421787 = r3421786 * r3421779;
        double r3421788 = r3421785 / r3421787;
        return r3421788;
}
↓
double f(double a, double b, double c) {
        double r3421789 = b;
        double r3421790 = -6239046376.848015;
        bool r3421791 = r3421789 <= r3421790;
        double r3421792 = c;
        double r3421793 = -r3421792;
        double r3421794 = r3421793 / r3421789;
        double r3421795 = -2.396838732613953e-306;
        bool r3421796 = r3421789 <= r3421795;
        double r3421797 = 0.5;
        double r3421798 = a;
        double r3421799 = r3421797 / r3421798;
        double r3421800 = 4.0;
        double r3421801 = r3421792 * r3421800;
        double r3421802 = r3421801 * r3421798;
        double r3421803 = r3421799 * r3421802;
        double r3421804 = -r3421789;
        double r3421805 = r3421789 * r3421789;
        double r3421806 = r3421792 * r3421798;
        double r3421807 = r3421800 * r3421806;
        double r3421808 = r3421805 - r3421807;
        double r3421809 = sqrt(r3421808);
        double r3421810 = r3421804 + r3421809;
        double r3421811 = r3421803 / r3421810;
        double r3421812 = 9.179168538250646e+63;
        bool r3421813 = r3421789 <= r3421812;
        double r3421814 = r3421804 - r3421809;
        double r3421815 = r3421799 * r3421814;
        double r3421816 = r3421804 / r3421798;
        double r3421817 = r3421813 ? r3421815 : r3421816;
        double r3421818 = r3421796 ? r3421811 : r3421817;
        double r3421819 = r3421791 ? r3421794 : r3421818;
        return r3421819;
}

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

Original33.7
Target20.7
Herbie9.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -6239046376.848015

    1. Initial program 55.7

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

    2. Taylor expanded around -inf 5.1

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

    3. Simplified5.1

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

    if -6239046376.848015 < b < -2.396838732613953e-306

    1. Initial program 27.2

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

    3. Applied clear-num27.2

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

    5. Applied flip--27.3

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

    6. Applied associate-/r/27.3

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

    7. Applied associate-/r*27.3

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

    8. Simplified17.2

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

    if -2.396838732613953e-306 < b < 9.179168538250646e+63

    1. Initial program 9.1

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

    3. Applied clear-num9.3

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

    5. Applied associate-/r/9.3

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

    6. Simplified9.3

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

    if 9.179168538250646e+63 < b

    1. Initial program 37.8

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

    3. Applied clear-num37.9

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

    4. Taylor expanded around 0 6.1

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

    5. Simplified6.1

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

  4. Final simplification9.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6239046376.848015:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -2.396838732613953 \cdot 10^{-306}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{a} \cdot \left(\left(c \cdot 4\right) \cdot a\right)}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\\ \mathbf{elif}\;b \le 9.179168538250646 \cdot 10^{+63}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))