Average Error: 34.4 → 10.3
Time: 21.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, 2, -2 \cdot \frac{b}{a}\right)}{2}\\ \mathbf{elif}\;b \le 9.136492990928292133394320076175633285536 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]
\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 -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, 2, -2 \cdot \frac{b}{a}\right)}{2}\\

\mathbf{elif}\;b \le 9.136492990928292133394320076175633285536 \cdot 10^{-23}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)} - b\right)}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3060957 = b;
        double r3060958 = -r3060957;
        double r3060959 = r3060957 * r3060957;
        double r3060960 = 4.0;
        double r3060961 = a;
        double r3060962 = c;
        double r3060963 = r3060961 * r3060962;
        double r3060964 = r3060960 * r3060963;
        double r3060965 = r3060959 - r3060964;
        double r3060966 = sqrt(r3060965);
        double r3060967 = r3060958 + r3060966;
        double r3060968 = 2.0;
        double r3060969 = r3060968 * r3060961;
        double r3060970 = r3060967 / r3060969;
        return r3060970;
}
↓
double f(double a, double b, double c) {
        double r3060971 = b;
        double r3060972 = -1.7633154797394035e+89;
        bool r3060973 = r3060971 <= r3060972;
        double r3060974 = c;
        double r3060975 = r3060974 / r3060971;
        double r3060976 = 2.0;
        double r3060977 = -2.0;
        double r3060978 = a;
        double r3060979 = r3060971 / r3060978;
        double r3060980 = r3060977 * r3060979;
        double r3060981 = fma(r3060975, r3060976, r3060980);
        double r3060982 = r3060981 / r3060976;
        double r3060983 = 9.136492990928292e-23;
        bool r3060984 = r3060971 <= r3060983;
        double r3060985 = 1.0;
        double r3060986 = r3060985 / r3060978;
        double r3060987 = r3060971 * r3060971;
        double r3060988 = 4.0;
        double r3060989 = r3060988 * r3060974;
        double r3060990 = r3060978 * r3060989;
        double r3060991 = r3060987 - r3060990;
        double r3060992 = sqrt(r3060991);
        double r3060993 = r3060992 - r3060971;
        double r3060994 = r3060986 * r3060993;
        double r3060995 = r3060994 / r3060976;
        double r3060996 = -2.0;
        double r3060997 = r3060975 * r3060996;
        double r3060998 = r3060997 / r3060976;
        double r3060999 = r3060984 ? r3060995 : r3060998;
        double r3061000 = r3060973 ? r3060982 : r3060999;
        return r3061000;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.4
Target21.3
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.7633154797394035e+89

    1. Initial program 45.7

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

    2. Simplified45.7

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

    4. Applied div-inv45.8

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

    5. Taylor expanded around -inf 3.9

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

    6. Simplified3.9

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

    if -1.7633154797394035e+89 < b < 9.136492990928292e-23

    1. Initial program 15.0

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

    2. Simplified15.1

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

    4. Applied div-inv15.2

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

    if 9.136492990928292e-23 < b

    1. Initial program 55.4

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

    2. Simplified55.4

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

    3. Taylor expanded around inf 6.7

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, 2, -2 \cdot \frac{b}{a}\right)}{2}\\ \mathbf{elif}\;b \le 9.136492990928292133394320076175633285536 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

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

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