Average Error: 33.3 → 10.5
Time: 20.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.153478880637207 \cdot 10^{+108}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.8378252714625124 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(\left(a \cdot -4\right) \cdot c\right)\right)} - b}{2}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -1.153478880637207 \cdot 10^{+108}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 1.8378252714625124 \cdot 10^{-19}:\\
\;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(\left(a \cdot -4\right) \cdot c\right)\right)} - b}{2}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2322896 = b;
        double r2322897 = -r2322896;
        double r2322898 = r2322896 * r2322896;
        double r2322899 = 4.0;
        double r2322900 = a;
        double r2322901 = r2322899 * r2322900;
        double r2322902 = c;
        double r2322903 = r2322901 * r2322902;
        double r2322904 = r2322898 - r2322903;
        double r2322905 = sqrt(r2322904);
        double r2322906 = r2322897 + r2322905;
        double r2322907 = 2.0;
        double r2322908 = r2322907 * r2322900;
        double r2322909 = r2322906 / r2322908;
        return r2322909;
}
↓
double f(double a, double b, double c) {
        double r2322910 = b;
        double r2322911 = -1.153478880637207e+108;
        bool r2322912 = r2322910 <= r2322911;
        double r2322913 = c;
        double r2322914 = r2322913 / r2322910;
        double r2322915 = a;
        double r2322916 = r2322910 / r2322915;
        double r2322917 = r2322914 - r2322916;
        double r2322918 = 1.8378252714625124e-19;
        bool r2322919 = r2322910 <= r2322918;
        double r2322920 = 1.0;
        double r2322921 = -4.0;
        double r2322922 = r2322915 * r2322921;
        double r2322923 = r2322922 * r2322913;
        double r2322924 = fma(r2322910, r2322910, r2322923);
        double r2322925 = sqrt(r2322924);
        double r2322926 = r2322925 - r2322910;
        double r2322927 = 2.0;
        double r2322928 = r2322926 / r2322927;
        double r2322929 = r2322915 / r2322928;
        double r2322930 = r2322920 / r2322929;
        double r2322931 = -r2322914;
        double r2322932 = r2322919 ? r2322930 : r2322931;
        double r2322933 = r2322912 ? r2322917 : r2322932;
        return r2322933;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.3
Target20.3
Herbie10.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.153478880637207e+108

    1. Initial program 46.3

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

    2. Simplified46.3

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

    3. Taylor expanded around -inf 3.2

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

    if -1.153478880637207e+108 < b < 1.8378252714625124e-19

    1. Initial program 14.9

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

    2. Simplified14.8

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

    4. Applied clear-num15.0

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

    if 1.8378252714625124e-19 < b

    1. Initial program 54.4

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

    2. Simplified54.4

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

    4. Applied *-un-lft-identity54.4

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

    5. Applied div-inv54.4

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

    6. Applied times-frac54.4

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

    7. Simplified54.4

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

    8. Simplified54.4

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

    9. Taylor expanded around inf 7.0

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

    10. Simplified7.0

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.153478880637207 \cdot 10^{+108}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.8378252714625124 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(\left(a \cdot -4\right) \cdot c\right)\right)} - b}{2}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(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)))