Average Error: 33.6 → 9.8
Time: 23.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.694684309811035 \cdot 10^{+121}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 4.6659701943749105 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -4.694684309811035 \cdot 10^{+121}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 4.6659701943749105 \cdot 10^{-84}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b\right)}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1458974 = b;
        double r1458975 = -r1458974;
        double r1458976 = r1458974 * r1458974;
        double r1458977 = 4.0;
        double r1458978 = a;
        double r1458979 = r1458977 * r1458978;
        double r1458980 = c;
        double r1458981 = r1458979 * r1458980;
        double r1458982 = r1458976 - r1458981;
        double r1458983 = sqrt(r1458982);
        double r1458984 = r1458975 + r1458983;
        double r1458985 = 2.0;
        double r1458986 = r1458985 * r1458978;
        double r1458987 = r1458984 / r1458986;
        return r1458987;
}
↓
double f(double a, double b, double c) {
        double r1458988 = b;
        double r1458989 = -4.694684309811035e+121;
        bool r1458990 = r1458988 <= r1458989;
        double r1458991 = c;
        double r1458992 = r1458991 / r1458988;
        double r1458993 = a;
        double r1458994 = r1458988 / r1458993;
        double r1458995 = r1458992 - r1458994;
        double r1458996 = 2.0;
        double r1458997 = r1458995 * r1458996;
        double r1458998 = r1458997 / r1458996;
        double r1458999 = 4.6659701943749105e-84;
        bool r1459000 = r1458988 <= r1458999;
        double r1459001 = 1.0;
        double r1459002 = r1459001 / r1458993;
        double r1459003 = -4.0;
        double r1459004 = r1458993 * r1459003;
        double r1459005 = r1458988 * r1458988;
        double r1459006 = fma(r1458991, r1459004, r1459005);
        double r1459007 = sqrt(r1459006);
        double r1459008 = r1459007 - r1458988;
        double r1459009 = r1459002 * r1459008;
        double r1459010 = r1459009 / r1458996;
        double r1459011 = -2.0;
        double r1459012 = r1459011 * r1458992;
        double r1459013 = r1459012 / r1458996;
        double r1459014 = r1459000 ? r1459010 : r1459013;
        double r1459015 = r1458990 ? r1458998 : r1459014;
        return r1459015;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.6
Target20.4
Herbie9.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.694684309811035e+121

    1. Initial program 49.8

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

    2. Simplified49.8

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

    3. Taylor expanded around -inf 2.6

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

    4. Simplified2.6

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

    if -4.694684309811035e+121 < b < 4.6659701943749105e-84

    1. Initial program 12.2

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

    2. Simplified12.2

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

    3. Taylor expanded around -inf 12.2

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

    4. Simplified12.2

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

    6. Applied div-inv12.3

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

    if 4.6659701943749105e-84 < b

    1. Initial program 52.2

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

    2. Simplified52.2

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

    3. Taylor expanded around inf 9.3

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.694684309811035 \cdot 10^{+121}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 4.6659701943749105 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 +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)))