Average Error: 33.4 → 9.9
Time: 28.5s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.840085388791461 \cdot 10^{-68}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.5949594684703287 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(\frac{c}{\frac{b}{a}} - b\right)}{2}}{a}\\ \end{array}\]
\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 -2.840085388791461 \cdot 10^{-68}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 1.5949594684703287 \cdot 10^{+126}:\\
\;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)}}{2}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{2 \cdot \left(\frac{c}{\frac{b}{a}} - b\right)}{2}}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r2281984 = b;
        double r2281985 = -r2281984;
        double r2281986 = r2281984 * r2281984;
        double r2281987 = 4.0;
        double r2281988 = a;
        double r2281989 = c;
        double r2281990 = r2281988 * r2281989;
        double r2281991 = r2281987 * r2281990;
        double r2281992 = r2281986 - r2281991;
        double r2281993 = sqrt(r2281992);
        double r2281994 = r2281985 - r2281993;
        double r2281995 = 2.0;
        double r2281996 = r2281995 * r2281988;
        double r2281997 = r2281994 / r2281996;
        return r2281997;
}
↓
double f(double a, double b, double c) {
        double r2281998 = b;
        double r2281999 = -2.840085388791461e-68;
        bool r2282000 = r2281998 <= r2281999;
        double r2282001 = c;
        double r2282002 = r2282001 / r2281998;
        double r2282003 = -r2282002;
        double r2282004 = 1.5949594684703287e+126;
        bool r2282005 = r2281998 <= r2282004;
        double r2282006 = -r2281998;
        double r2282007 = -4.0;
        double r2282008 = a;
        double r2282009 = r2282008 * r2282001;
        double r2282010 = r2281998 * r2281998;
        double r2282011 = fma(r2282007, r2282009, r2282010);
        double r2282012 = sqrt(r2282011);
        double r2282013 = r2282006 - r2282012;
        double r2282014 = 2.0;
        double r2282015 = r2282013 / r2282014;
        double r2282016 = r2282015 / r2282008;
        double r2282017 = r2281998 / r2282008;
        double r2282018 = r2282001 / r2282017;
        double r2282019 = r2282018 - r2281998;
        double r2282020 = r2282014 * r2282019;
        double r2282021 = r2282020 / r2282014;
        double r2282022 = r2282021 / r2282008;
        double r2282023 = r2282005 ? r2282016 : r2282022;
        double r2282024 = r2282000 ? r2282003 : r2282023;
        return r2282024;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.4
Target20.2
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.840085388791461e-68

    1. Initial program 52.7

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

    2. Simplified52.7

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

    3. Taylor expanded around inf 52.7

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

    4. Simplified52.7

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

    5. Taylor expanded around -inf 9.0

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

    6. Simplified9.0

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

    if -2.840085388791461e-68 < b < 1.5949594684703287e+126

    1. Initial program 12.5

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

    2. Simplified12.5

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

    3. Taylor expanded around inf 12.5

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

    4. Simplified12.5

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

    if 1.5949594684703287e+126 < b

    1. Initial program 50.8

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

    2. Simplified50.8

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

    3. Taylor expanded around inf 10.6

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

    4. Simplified3.2

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.840085388791461 \cdot 10^{-68}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.5949594684703287 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, \left(a \cdot c\right), \left(b \cdot b\right)\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(\frac{c}{\frac{b}{a}} - b\right)}{2}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 +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)))