Average Error: 34.2 → 11.6
Time: 19.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.555632367828988861043913196266489993904 \cdot 10^{101}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.555632367828988861043913196266489993904 \cdot 10^{101}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r56793 = b;
        double r56794 = -r56793;
        double r56795 = r56793 * r56793;
        double r56796 = 4.0;
        double r56797 = a;
        double r56798 = r56796 * r56797;
        double r56799 = c;
        double r56800 = r56798 * r56799;
        double r56801 = r56795 - r56800;
        double r56802 = sqrt(r56801);
        double r56803 = r56794 + r56802;
        double r56804 = 2.0;
        double r56805 = r56804 * r56797;
        double r56806 = r56803 / r56805;
        return r56806;
}
↓
double f(double a, double b, double c) {
        double r56807 = b;
        double r56808 = -1.555632367828989e+101;
        bool r56809 = r56807 <= r56808;
        double r56810 = 1.0;
        double r56811 = c;
        double r56812 = r56811 / r56807;
        double r56813 = a;
        double r56814 = r56807 / r56813;
        double r56815 = r56812 - r56814;
        double r56816 = r56810 * r56815;
        double r56817 = 7.455592343308264e-170;
        bool r56818 = r56807 <= r56817;
        double r56819 = -r56807;
        double r56820 = 2.0;
        double r56821 = pow(r56807, r56820);
        double r56822 = 4.0;
        double r56823 = r56813 * r56811;
        double r56824 = r56822 * r56823;
        double r56825 = r56821 - r56824;
        double r56826 = sqrt(r56825);
        double r56827 = r56819 + r56826;
        double r56828 = 2.0;
        double r56829 = r56828 * r56813;
        double r56830 = r56827 / r56829;
        double r56831 = -1.0;
        double r56832 = r56831 * r56812;
        double r56833 = r56818 ? r56830 : r56832;
        double r56834 = r56809 ? r56816 : r56833;
        return r56834;
}

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

Original34.2
Target20.8
Herbie11.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.555632367828989e+101

    1. Initial program 47.4

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

    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 -1.555632367828989e+101 < b < 7.455592343308264e-170

    1. Initial program 11.7

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

    2. Taylor expanded around 0 11.7

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

    if 7.455592343308264e-170 < b

    1. Initial program 48.9

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

    2. Taylor expanded around inf 14.1

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

  4. Final simplification11.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.555632367828988861043913196266489993904 \cdot 10^{101}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :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)))