Average Error: 33.2 → 9.7
Time: 24.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.170773079316174 \cdot 10^{+99}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 3.0168583404714427 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -4.170773079316174 \cdot 10^{+99}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 3.0168583404714427 \cdot 10^{-66}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a} - \frac{b}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r3995818 = b;
        double r3995819 = -r3995818;
        double r3995820 = r3995818 * r3995818;
        double r3995821 = 4.0;
        double r3995822 = a;
        double r3995823 = r3995821 * r3995822;
        double r3995824 = c;
        double r3995825 = r3995823 * r3995824;
        double r3995826 = r3995820 - r3995825;
        double r3995827 = sqrt(r3995826);
        double r3995828 = r3995819 + r3995827;
        double r3995829 = 2.0;
        double r3995830 = r3995829 * r3995822;
        double r3995831 = r3995828 / r3995830;
        return r3995831;
}
↓
double f(double a, double b, double c) {
        double r3995832 = b;
        double r3995833 = -4.170773079316174e+99;
        bool r3995834 = r3995832 <= r3995833;
        double r3995835 = c;
        double r3995836 = r3995835 / r3995832;
        double r3995837 = a;
        double r3995838 = r3995832 / r3995837;
        double r3995839 = r3995836 - r3995838;
        double r3995840 = 2.0;
        double r3995841 = r3995839 * r3995840;
        double r3995842 = r3995841 / r3995840;
        double r3995843 = 3.0168583404714427e-66;
        bool r3995844 = r3995832 <= r3995843;
        double r3995845 = r3995832 * r3995832;
        double r3995846 = 4.0;
        double r3995847 = r3995835 * r3995837;
        double r3995848 = r3995846 * r3995847;
        double r3995849 = r3995845 - r3995848;
        double r3995850 = sqrt(r3995849);
        double r3995851 = r3995850 / r3995837;
        double r3995852 = r3995851 - r3995838;
        double r3995853 = r3995852 / r3995840;
        double r3995854 = -2.0;
        double r3995855 = r3995836 * r3995854;
        double r3995856 = r3995855 / r3995840;
        double r3995857 = r3995844 ? r3995853 : r3995856;
        double r3995858 = r3995834 ? r3995842 : r3995857;
        return r3995858;
}

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

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

Derivation

  1. Split input into 3 regimes

  2. if b < -4.170773079316174e+99

    1. Initial program 44.2

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

    2. Simplified44.2

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

    4. Applied div-inv44.3

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

    5. Taylor expanded around -inf 3.3

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

    6. Simplified3.3

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

    if -4.170773079316174e+99 < b < 3.0168583404714427e-66

    1. Initial program 12.8

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

    2. Simplified12.8

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

    4. Applied div-sub12.8

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

    if 3.0168583404714427e-66 < b

    1. Initial program 53.1

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

    2. Simplified53.0

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

    4. Applied div-inv53.0

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

    5. Taylor expanded around inf 8.5

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.170773079316174 \cdot 10^{+99}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 3.0168583404714427 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 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)))