Average Error: 34.3 → 9.9
Time: 30.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.1113133556666892392847386011681009173 \cdot 10^{-81}:\\ \;\;\;\;-\frac{1 \cdot c}{b}\\ \mathbf{elif}\;b \le 3.583649041028097672662453906912522703022 \cdot 10^{84}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(a \cdot \left(-c\right), 4, b \cdot b\right)} + b\right) \cdot \frac{-1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b + b}{a \cdot 2}\\ \end{array}\]
\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 -5.1113133556666892392847386011681009173 \cdot 10^{-81}:\\
\;\;\;\;-\frac{1 \cdot c}{b}\\

\mathbf{elif}\;b \le 3.583649041028097672662453906912522703022 \cdot 10^{84}:\\
\;\;\;\;\left(\sqrt{\mathsf{fma}\left(a \cdot \left(-c\right), 4, b \cdot b\right)} + b\right) \cdot \frac{-1}{a \cdot 2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r48693 = b;
        double r48694 = -r48693;
        double r48695 = r48693 * r48693;
        double r48696 = 4.0;
        double r48697 = a;
        double r48698 = c;
        double r48699 = r48697 * r48698;
        double r48700 = r48696 * r48699;
        double r48701 = r48695 - r48700;
        double r48702 = sqrt(r48701);
        double r48703 = r48694 - r48702;
        double r48704 = 2.0;
        double r48705 = r48704 * r48697;
        double r48706 = r48703 / r48705;
        return r48706;
}
↓
double f(double a, double b, double c) {
        double r48707 = b;
        double r48708 = -5.111313355666689e-81;
        bool r48709 = r48707 <= r48708;
        double r48710 = 1.0;
        double r48711 = c;
        double r48712 = r48710 * r48711;
        double r48713 = r48712 / r48707;
        double r48714 = -r48713;
        double r48715 = 3.5836490410280977e+84;
        bool r48716 = r48707 <= r48715;
        double r48717 = a;
        double r48718 = -r48711;
        double r48719 = r48717 * r48718;
        double r48720 = 4.0;
        double r48721 = r48707 * r48707;
        double r48722 = fma(r48719, r48720, r48721);
        double r48723 = sqrt(r48722);
        double r48724 = r48723 + r48707;
        double r48725 = -1.0;
        double r48726 = 2.0;
        double r48727 = r48717 * r48726;
        double r48728 = r48725 / r48727;
        double r48729 = r48724 * r48728;
        double r48730 = r48707 + r48707;
        double r48731 = r48730 / r48727;
        double r48732 = -r48731;
        double r48733 = r48716 ? r48729 : r48732;
        double r48734 = r48709 ? r48714 : r48733;
        return r48734;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.3
Target21.1
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -5.111313355666689e-81

    1. Initial program 53.4

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

    2. Simplified53.4

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

    3. Taylor expanded around -inf 8.9

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

    4. Simplified9.6

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

    5. Taylor expanded around 0 8.9

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

    6. Simplified8.9

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

    if -5.111313355666689e-81 < b < 3.5836490410280977e+84

    1. Initial program 13.2

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

    2. Simplified13.2

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

    4. Applied div-inv13.3

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

    5. Simplified13.3

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

    if 3.5836490410280977e+84 < b

    1. Initial program 44.0

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

    2. Simplified44.0

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

    3. Taylor expanded around 0 3.7

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.1113133556666892392847386011681009173 \cdot 10^{-81}:\\ \;\;\;\;-\frac{1 \cdot c}{b}\\ \mathbf{elif}\;b \le 3.583649041028097672662453906912522703022 \cdot 10^{84}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(a \cdot \left(-c\right), 4, b \cdot b\right)} + b\right) \cdot \frac{-1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b + b}{a \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))