Average Error: 32.9 → 10.3
Time: 20.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.088000531423294 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 9.354082991670835 \cdot 10^{-125}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)} - b\right) \cdot \frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]
\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 -9.088000531423294 \cdot 10^{+152}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 9.354082991670835 \cdot 10^{-125}:\\
\;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)} - b\right) \cdot \frac{1}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1241704 = b;
        double r1241705 = -r1241704;
        double r1241706 = r1241704 * r1241704;
        double r1241707 = 4.0;
        double r1241708 = a;
        double r1241709 = r1241707 * r1241708;
        double r1241710 = c;
        double r1241711 = r1241709 * r1241710;
        double r1241712 = r1241706 - r1241711;
        double r1241713 = sqrt(r1241712);
        double r1241714 = r1241705 + r1241713;
        double r1241715 = 2.0;
        double r1241716 = r1241715 * r1241708;
        double r1241717 = r1241714 / r1241716;
        return r1241717;
}
↓
double f(double a, double b, double c) {
        double r1241718 = b;
        double r1241719 = -9.088000531423294e+152;
        bool r1241720 = r1241718 <= r1241719;
        double r1241721 = c;
        double r1241722 = r1241721 / r1241718;
        double r1241723 = a;
        double r1241724 = r1241718 / r1241723;
        double r1241725 = r1241722 - r1241724;
        double r1241726 = 9.354082991670835e-125;
        bool r1241727 = r1241718 <= r1241726;
        double r1241728 = -4.0;
        double r1241729 = r1241723 * r1241728;
        double r1241730 = r1241718 * r1241718;
        double r1241731 = fma(r1241729, r1241721, r1241730);
        double r1241732 = sqrt(r1241731);
        double r1241733 = r1241732 - r1241718;
        double r1241734 = 0.5;
        double r1241735 = r1241733 * r1241734;
        double r1241736 = r1241735 / r1241723;
        double r1241737 = -r1241721;
        double r1241738 = r1241737 / r1241718;
        double r1241739 = r1241727 ? r1241736 : r1241738;
        double r1241740 = r1241720 ? r1241725 : r1241739;
        return r1241740;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original32.9
Target20.3
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -9.088000531423294e+152

    1. Initial program 60.4

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

    2. Simplified60.4

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

    3. Taylor expanded around -inf 1.5

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

    if -9.088000531423294e+152 < b < 9.354082991670835e-125

    1. Initial program 10.9

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

    2. Simplified10.9

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

    4. Applied *-un-lft-identity10.9

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

    5. Applied div-inv11.1

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

    6. Applied times-frac11.1

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

    7. Simplified11.1

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

    8. Simplified11.1

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

    10. Applied associate-*r/10.9

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

    if 9.354082991670835e-125 < b

    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 11.9

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

    4. Simplified11.9

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.088000531423294 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 9.354082991670835 \cdot 10^{-125}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)} - b\right) \cdot \frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +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)))