Average Error: 32.8 → 10.0
Time: 20.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.063397748446981 \cdot 10^{+71}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 3.1295384133612364 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -3.063397748446981 \cdot 10^{+71}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 3.1295384133612364 \cdot 10^{-73}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b\right)}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2008759 = b;
        double r2008760 = -r2008759;
        double r2008761 = r2008759 * r2008759;
        double r2008762 = 4.0;
        double r2008763 = a;
        double r2008764 = r2008762 * r2008763;
        double r2008765 = c;
        double r2008766 = r2008764 * r2008765;
        double r2008767 = r2008761 - r2008766;
        double r2008768 = sqrt(r2008767);
        double r2008769 = r2008760 + r2008768;
        double r2008770 = 2.0;
        double r2008771 = r2008770 * r2008763;
        double r2008772 = r2008769 / r2008771;
        return r2008772;
}
↓
double f(double a, double b, double c) {
        double r2008773 = b;
        double r2008774 = -3.063397748446981e+71;
        bool r2008775 = r2008773 <= r2008774;
        double r2008776 = c;
        double r2008777 = r2008776 / r2008773;
        double r2008778 = a;
        double r2008779 = r2008773 / r2008778;
        double r2008780 = r2008777 - r2008779;
        double r2008781 = 2.0;
        double r2008782 = r2008780 * r2008781;
        double r2008783 = r2008782 / r2008781;
        double r2008784 = 3.1295384133612364e-73;
        bool r2008785 = r2008773 <= r2008784;
        double r2008786 = 1.0;
        double r2008787 = r2008786 / r2008778;
        double r2008788 = -4.0;
        double r2008789 = r2008788 * r2008778;
        double r2008790 = r2008789 * r2008776;
        double r2008791 = fma(r2008773, r2008773, r2008790);
        double r2008792 = sqrt(r2008791);
        double r2008793 = r2008792 - r2008773;
        double r2008794 = r2008787 * r2008793;
        double r2008795 = r2008794 / r2008781;
        double r2008796 = -2.0;
        double r2008797 = r2008796 * r2008777;
        double r2008798 = r2008797 / r2008781;
        double r2008799 = r2008785 ? r2008795 : r2008798;
        double r2008800 = r2008775 ? r2008783 : r2008799;
        return r2008800;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original32.8
Target20.1
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.063397748446981e+71

    1. Initial program 38.6

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

    2. Simplified38.6

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

    4. Applied div-inv38.7

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

    5. Taylor expanded around -inf 4.7

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

    6. Simplified4.7

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

    if -3.063397748446981e+71 < b < 3.1295384133612364e-73

    1. Initial program 13.0

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

    2. Simplified13.0

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

    4. Applied div-inv13.2

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

    if 3.1295384133612364e-73 < b

    1. Initial program 52.3

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

    2. Simplified52.3

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

    3. Taylor expanded around inf 9.0

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.063397748446981 \cdot 10^{+71}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 3.1295384133612364 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +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)))