Average Error: 33.5 → 10.6
Time: 3.7m
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.691277786452672 \cdot 10^{-38}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.8091015183831773 \cdot 10^{+43}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -5.691277786452672 \cdot 10^{-38}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 1.8091015183831773 \cdot 10^{+43}:\\
\;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r13317882 = b;
        double r13317883 = -r13317882;
        double r13317884 = r13317882 * r13317882;
        double r13317885 = 4.0;
        double r13317886 = a;
        double r13317887 = c;
        double r13317888 = r13317886 * r13317887;
        double r13317889 = r13317885 * r13317888;
        double r13317890 = r13317884 - r13317889;
        double r13317891 = sqrt(r13317890);
        double r13317892 = r13317883 - r13317891;
        double r13317893 = 2.0;
        double r13317894 = r13317893 * r13317886;
        double r13317895 = r13317892 / r13317894;
        return r13317895;
}
↓
double f(double a, double b, double c) {
        double r13317896 = b;
        double r13317897 = -5.691277786452672e-38;
        bool r13317898 = r13317896 <= r13317897;
        double r13317899 = c;
        double r13317900 = r13317899 / r13317896;
        double r13317901 = -r13317900;
        double r13317902 = 1.8091015183831773e+43;
        bool r13317903 = r13317896 <= r13317902;
        double r13317904 = -r13317896;
        double r13317905 = a;
        double r13317906 = r13317899 * r13317905;
        double r13317907 = -4.0;
        double r13317908 = r13317896 * r13317896;
        double r13317909 = fma(r13317906, r13317907, r13317908);
        double r13317910 = sqrt(r13317909);
        double r13317911 = r13317904 - r13317910;
        double r13317912 = 2.0;
        double r13317913 = r13317911 / r13317912;
        double r13317914 = r13317913 / r13317905;
        double r13317915 = r13317896 / r13317905;
        double r13317916 = r13317900 - r13317915;
        double r13317917 = r13317903 ? r13317914 : r13317916;
        double r13317918 = r13317898 ? r13317901 : r13317917;
        return r13317918;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.5
Target21.1
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -5.691277786452672e-38

    1. Initial program 54.0

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

    2. Simplified54.0

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

    3. Taylor expanded around -inf 7.9

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

    4. Simplified7.9

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

    if -5.691277786452672e-38 < b < 1.8091015183831773e+43

    1. Initial program 15.3

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

    2. Simplified15.3

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

    3. Taylor expanded around -inf 15.3

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

    4. Simplified15.3

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

    if 1.8091015183831773e+43 < b

    1. Initial program 36.3

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

    2. Simplified36.3

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

    3. Taylor expanded around -inf 36.3

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

    4. Simplified36.3

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

    5. Taylor expanded around inf 5.7

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.691277786452672 \cdot 10^{-38}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.8091015183831773 \cdot 10^{+43}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(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)))