Average Error: 33.7 → 10.9
Time: 11.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.98276540088900058 \cdot 10^{134}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.1860189201379418 \cdot 10^{-161}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)} - b\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.98276540088900058 \cdot 10^{134}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.1860189201379418 \cdot 10^{-161}:\\
\;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)} - b\right) \cdot \frac{\frac{1}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r159886 = b;
        double r159887 = -r159886;
        double r159888 = r159886 * r159886;
        double r159889 = 4.0;
        double r159890 = a;
        double r159891 = r159889 * r159890;
        double r159892 = c;
        double r159893 = r159891 * r159892;
        double r159894 = r159888 - r159893;
        double r159895 = sqrt(r159894);
        double r159896 = r159887 + r159895;
        double r159897 = 2.0;
        double r159898 = r159897 * r159890;
        double r159899 = r159896 / r159898;
        return r159899;
}
↓
double f(double a, double b, double c) {
        double r159900 = b;
        double r159901 = -1.9827654008890006e+134;
        bool r159902 = r159900 <= r159901;
        double r159903 = 1.0;
        double r159904 = c;
        double r159905 = r159904 / r159900;
        double r159906 = a;
        double r159907 = r159900 / r159906;
        double r159908 = r159905 - r159907;
        double r159909 = r159903 * r159908;
        double r159910 = 1.1860189201379418e-161;
        bool r159911 = r159900 <= r159910;
        double r159912 = 4.0;
        double r159913 = r159912 * r159906;
        double r159914 = r159913 * r159904;
        double r159915 = -r159914;
        double r159916 = fma(r159900, r159900, r159915);
        double r159917 = sqrt(r159916);
        double r159918 = r159917 - r159900;
        double r159919 = 1.0;
        double r159920 = 2.0;
        double r159921 = r159919 / r159920;
        double r159922 = r159921 / r159906;
        double r159923 = r159918 * r159922;
        double r159924 = -1.0;
        double r159925 = r159924 * r159905;
        double r159926 = r159911 ? r159923 : r159925;
        double r159927 = r159902 ? r159909 : r159926;
        return r159927;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.7
Target21.0
Herbie10.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.9827654008890006e+134

    1. Initial program 56.8

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

    2. Simplified56.8

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

    3. Taylor expanded around -inf 3.1

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

    4. Simplified3.1

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

    if -1.9827654008890006e+134 < b < 1.1860189201379418e-161

    1. Initial program 10.3

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

    2. Simplified10.3

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

    4. Applied *-un-lft-identity10.3

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

    5. Applied div-inv10.3

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

    6. Applied times-frac10.5

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

    7. Simplified10.5

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

    if 1.1860189201379418e-161 < b

    1. Initial program 49.7

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

    2. Simplified49.7

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

    3. Taylor expanded around inf 13.7

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

  4. Final simplification10.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.98276540088900058 \cdot 10^{134}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.1860189201379418 \cdot 10^{-161}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)} - b\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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)))