Average Error: 33.4 → 9.8
Time: 28.4s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.852138444177435 \cdot 10^{-54}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 6.359263193477048 \cdot 10^{+137}:\\ \;\;\;\;\frac{1}{a \cdot \frac{2}{-\left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)} + b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -2.852138444177435 \cdot 10^{-54}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 6.359263193477048 \cdot 10^{+137}:\\
\;\;\;\;\frac{1}{a \cdot \frac{2}{-\left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)} + b\right)}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2390829 = b;
        double r2390830 = -r2390829;
        double r2390831 = r2390829 * r2390829;
        double r2390832 = 4.0;
        double r2390833 = a;
        double r2390834 = c;
        double r2390835 = r2390833 * r2390834;
        double r2390836 = r2390832 * r2390835;
        double r2390837 = r2390831 - r2390836;
        double r2390838 = sqrt(r2390837);
        double r2390839 = r2390830 - r2390838;
        double r2390840 = 2.0;
        double r2390841 = r2390840 * r2390833;
        double r2390842 = r2390839 / r2390841;
        return r2390842;
}
↓
double f(double a, double b, double c) {
        double r2390843 = b;
        double r2390844 = -2.852138444177435e-54;
        bool r2390845 = r2390843 <= r2390844;
        double r2390846 = c;
        double r2390847 = r2390846 / r2390843;
        double r2390848 = -r2390847;
        double r2390849 = 6.359263193477048e+137;
        bool r2390850 = r2390843 <= r2390849;
        double r2390851 = 1.0;
        double r2390852 = a;
        double r2390853 = 2.0;
        double r2390854 = r2390846 * r2390852;
        double r2390855 = -4.0;
        double r2390856 = r2390854 * r2390855;
        double r2390857 = fma(r2390843, r2390843, r2390856);
        double r2390858 = sqrt(r2390857);
        double r2390859 = r2390858 + r2390843;
        double r2390860 = -r2390859;
        double r2390861 = r2390853 / r2390860;
        double r2390862 = r2390852 * r2390861;
        double r2390863 = r2390851 / r2390862;
        double r2390864 = r2390843 / r2390852;
        double r2390865 = r2390847 - r2390864;
        double r2390866 = r2390850 ? r2390863 : r2390865;
        double r2390867 = r2390845 ? r2390848 : r2390866;
        return r2390867;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.4
Target20.8
Herbie9.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.852138444177435e-54

    1. Initial program 53.4

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

    3. Applied div-inv53.4

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

    4. Simplified53.4

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

    5. Taylor expanded around -inf 8.3

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

    6. Simplified8.3

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

    if -2.852138444177435e-54 < b < 6.359263193477048e+137

    1. Initial program 12.6

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

    3. Applied *-un-lft-identity12.6

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

    4. Applied *-un-lft-identity12.6

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

    5. Applied distribute-rgt-neg-in12.6

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

    6. Applied distribute-lft-out--12.6

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

    7. Applied associate-/l*12.7

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

    8. Simplified12.8

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

    if 6.359263193477048e+137 < b

    1. Initial program 53.0

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

    2. Taylor expanded around inf 2.5

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.852138444177435 \cdot 10^{-54}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 6.359263193477048 \cdot 10^{+137}:\\ \;\;\;\;\frac{1}{a \cdot \frac{2}{-\left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)} + b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 +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)))