Average Error: 34.0 → 12.1
Time: 26.0s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.544460916074322 \cdot 10^{-48}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 1.5983000936606613 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{\frac{b \cdot b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} + \left(-b\right)}}{a}}{2}\\ \mathbf{elif}\;b \le 2.6656023684116586 \cdot 10^{+55}:\\ \;\;\;\;\frac{\frac{-\frac{\mathsf{fma}\left(b \cdot b, b, \mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}\right)}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}, \sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)} - b, b \cdot b\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{a} \cdot -2}{2}\\ \end{array}\]
\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 -8.544460916074322 \cdot 10^{-48}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\mathbf{elif}\;b \le 1.5983000936606613 \cdot 10^{-121}:\\
\;\;\;\;\frac{\frac{\frac{b \cdot b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} + \left(-b\right)}}{a}}{2}\\

\mathbf{elif}\;b \le 2.6656023684116586 \cdot 10^{+55}:\\
\;\;\;\;\frac{\frac{-\frac{\mathsf{fma}\left(b \cdot b, b, \mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}\right)}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}, \sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)} - b, b \cdot b\right)}}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2059788 = b;
        double r2059789 = -r2059788;
        double r2059790 = r2059788 * r2059788;
        double r2059791 = 4.0;
        double r2059792 = a;
        double r2059793 = c;
        double r2059794 = r2059792 * r2059793;
        double r2059795 = r2059791 * r2059794;
        double r2059796 = r2059790 - r2059795;
        double r2059797 = sqrt(r2059796);
        double r2059798 = r2059789 - r2059797;
        double r2059799 = 2.0;
        double r2059800 = r2059799 * r2059792;
        double r2059801 = r2059798 / r2059800;
        return r2059801;
}
↓
double f(double a, double b, double c) {
        double r2059802 = b;
        double r2059803 = -8.544460916074322e-48;
        bool r2059804 = r2059802 <= r2059803;
        double r2059805 = -2.0;
        double r2059806 = c;
        double r2059807 = r2059806 / r2059802;
        double r2059808 = r2059805 * r2059807;
        double r2059809 = 2.0;
        double r2059810 = r2059808 / r2059809;
        double r2059811 = 1.5983000936606613e-121;
        bool r2059812 = r2059802 <= r2059811;
        double r2059813 = r2059802 * r2059802;
        double r2059814 = a;
        double r2059815 = -4.0;
        double r2059816 = r2059814 * r2059815;
        double r2059817 = fma(r2059806, r2059816, r2059813);
        double r2059818 = sqrt(r2059817);
        double r2059819 = r2059818 * r2059818;
        double r2059820 = r2059813 - r2059819;
        double r2059821 = -r2059802;
        double r2059822 = r2059818 + r2059821;
        double r2059823 = r2059820 / r2059822;
        double r2059824 = r2059823 / r2059814;
        double r2059825 = r2059824 / r2059809;
        double r2059826 = 2.6656023684116586e+55;
        bool r2059827 = r2059802 <= r2059826;
        double r2059828 = r2059815 * r2059806;
        double r2059829 = fma(r2059814, r2059828, r2059813);
        double r2059830 = sqrt(r2059829);
        double r2059831 = r2059829 * r2059830;
        double r2059832 = fma(r2059813, r2059802, r2059831);
        double r2059833 = r2059830 - r2059802;
        double r2059834 = fma(r2059830, r2059833, r2059813);
        double r2059835 = r2059832 / r2059834;
        double r2059836 = -r2059835;
        double r2059837 = r2059836 / r2059814;
        double r2059838 = r2059837 / r2059809;
        double r2059839 = r2059802 / r2059814;
        double r2059840 = r2059839 * r2059805;
        double r2059841 = r2059840 / r2059809;
        double r2059842 = r2059827 ? r2059838 : r2059841;
        double r2059843 = r2059812 ? r2059825 : r2059842;
        double r2059844 = r2059804 ? r2059810 : r2059843;
        return r2059844;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.0
Target21.3
Herbie12.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -8.544460916074322e-48

    1. Initial program 54.0

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

    2. Simplified54.1

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

    3. Taylor expanded around -inf 7.8

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

    if -8.544460916074322e-48 < b < 1.5983000936606613e-121

    1. Initial program 19.2

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

    2. Simplified19.2

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

    4. Applied *-un-lft-identity19.2

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

    5. Applied *-un-lft-identity19.2

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

    6. Applied distribute-rgt-neg-in19.2

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

    7. Applied distribute-lft-out--19.2

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

    8. Applied associate-/l*19.3

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

    10. Applied div-inv19.3

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

    11. Simplified19.2

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

    13. Applied flip--21.1

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

    if 1.5983000936606613e-121 < b < 2.6656023684116586e+55

    1. Initial program 5.8

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

    2. Simplified5.9

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

    4. Applied flip3--12.9

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

    5. Simplified12.8

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

    6. Simplified12.8

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

    if 2.6656023684116586e+55 < b

    1. Initial program 37.1

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

    2. Simplified37.1

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

    4. Applied *-un-lft-identity37.1

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

    5. Applied *-un-lft-identity37.1

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

    6. Applied distribute-rgt-neg-in37.1

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

    7. Applied distribute-lft-out--37.1

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

    8. Applied associate-/l*37.2

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

    9. Taylor expanded around 0 6.5

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{b}{a}}}{2}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification12.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.544460916074322 \cdot 10^{-48}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 1.5983000936606613 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{\frac{b \cdot b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} + \left(-b\right)}}{a}}{2}\\ \mathbf{elif}\;b \le 2.6656023684116586 \cdot 10^{+55}:\\ \;\;\;\;\frac{\frac{-\frac{\mathsf{fma}\left(b \cdot b, b, \mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}\right)}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}, \sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)} - b, b \cdot b\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{a} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :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)))