Average Error: 32.5 → 9.8
Time: 41.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.4266250849096228 \cdot 10^{-56}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.2373425340727037 \cdot 10^{+98}:\\ \;\;\;\;\frac{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -1.4266250849096228 \cdot 10^{-56}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 2.2373425340727037 \cdot 10^{+98}:\\
\;\;\;\;\frac{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3382884 = b;
        double r3382885 = -r3382884;
        double r3382886 = r3382884 * r3382884;
        double r3382887 = 4.0;
        double r3382888 = a;
        double r3382889 = c;
        double r3382890 = r3382888 * r3382889;
        double r3382891 = r3382887 * r3382890;
        double r3382892 = r3382886 - r3382891;
        double r3382893 = sqrt(r3382892);
        double r3382894 = r3382885 - r3382893;
        double r3382895 = 2.0;
        double r3382896 = r3382895 * r3382888;
        double r3382897 = r3382894 / r3382896;
        return r3382897;
}
↓
double f(double a, double b, double c) {
        double r3382898 = b;
        double r3382899 = -1.4266250849096228e-56;
        bool r3382900 = r3382898 <= r3382899;
        double r3382901 = c;
        double r3382902 = r3382901 / r3382898;
        double r3382903 = -r3382902;
        double r3382904 = 2.2373425340727037e+98;
        bool r3382905 = r3382898 <= r3382904;
        double r3382906 = -r3382898;
        double r3382907 = -4.0;
        double r3382908 = a;
        double r3382909 = r3382908 * r3382901;
        double r3382910 = r3382898 * r3382898;
        double r3382911 = fma(r3382907, r3382909, r3382910);
        double r3382912 = sqrt(r3382911);
        double r3382913 = r3382906 - r3382912;
        double r3382914 = 0.5;
        double r3382915 = r3382913 * r3382914;
        double r3382916 = r3382915 / r3382908;
        double r3382917 = r3382898 / r3382908;
        double r3382918 = r3382902 - r3382917;
        double r3382919 = r3382905 ? r3382916 : r3382918;
        double r3382920 = r3382900 ? r3382903 : r3382919;
        return r3382920;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original32.5
Target20.0
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 < -1.4266250849096228e-56

    1. Initial program 52.5

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

    3. Applied div-inv52.5

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

    4. Simplified52.5

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

    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 -1.4266250849096228e-56 < b < 2.2373425340727037e+98

    1. Initial program 12.7

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

    3. Applied div-inv12.8

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

    4. Simplified12.8

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

    6. Applied associate-*r/12.7

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

    7. Simplified12.7

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

    if 2.2373425340727037e+98 < b

    1. Initial program 43.2

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

    3. Applied div-inv43.3

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

    4. Simplified43.3

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

    5. Taylor expanded around inf 4.9

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.4266250849096228 \cdot 10^{-56}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.2373425340727037 \cdot 10^{+98}:\\ \;\;\;\;\frac{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\right) \cdot \frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 +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)))