Average Error: 33.8 → 10.4
Time: 21.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.213216247196925388401125773743990732555 \cdot 10^{129}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{a}, -2, 2 \cdot \frac{c}{b}\right)}{2}\\ \mathbf{elif}\;b \le 6.088267304256603437292930310963869002155 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)} - b}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -8.213216247196925388401125773743990732555 \cdot 10^{129}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{a}, -2, 2 \cdot \frac{c}{b}\right)}{2}\\

\mathbf{elif}\;b \le 6.088267304256603437292930310963869002155 \cdot 10^{-81}:\\
\;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)} - b}}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3743866 = b;
        double r3743867 = -r3743866;
        double r3743868 = r3743866 * r3743866;
        double r3743869 = 4.0;
        double r3743870 = a;
        double r3743871 = r3743869 * r3743870;
        double r3743872 = c;
        double r3743873 = r3743871 * r3743872;
        double r3743874 = r3743868 - r3743873;
        double r3743875 = sqrt(r3743874);
        double r3743876 = r3743867 + r3743875;
        double r3743877 = 2.0;
        double r3743878 = r3743877 * r3743870;
        double r3743879 = r3743876 / r3743878;
        return r3743879;
}
↓
double f(double a, double b, double c) {
        double r3743880 = b;
        double r3743881 = -8.213216247196925e+129;
        bool r3743882 = r3743880 <= r3743881;
        double r3743883 = a;
        double r3743884 = r3743880 / r3743883;
        double r3743885 = -2.0;
        double r3743886 = 2.0;
        double r3743887 = c;
        double r3743888 = r3743887 / r3743880;
        double r3743889 = r3743886 * r3743888;
        double r3743890 = fma(r3743884, r3743885, r3743889);
        double r3743891 = r3743890 / r3743886;
        double r3743892 = 6.088267304256603e-81;
        bool r3743893 = r3743880 <= r3743892;
        double r3743894 = 1.0;
        double r3743895 = r3743880 * r3743880;
        double r3743896 = 4.0;
        double r3743897 = r3743887 * r3743896;
        double r3743898 = r3743883 * r3743897;
        double r3743899 = r3743895 - r3743898;
        double r3743900 = sqrt(r3743899);
        double r3743901 = r3743900 - r3743880;
        double r3743902 = r3743883 / r3743901;
        double r3743903 = r3743894 / r3743902;
        double r3743904 = r3743903 / r3743886;
        double r3743905 = -2.0;
        double r3743906 = r3743888 * r3743905;
        double r3743907 = r3743906 / r3743886;
        double r3743908 = r3743893 ? r3743904 : r3743907;
        double r3743909 = r3743882 ? r3743891 : r3743908;
        return r3743909;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.8
Target20.6
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -8.213216247196925e+129

    1. Initial program 53.9

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

    2. Simplified53.9

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

    3. Taylor expanded around -inf 2.5

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

    4. Simplified2.5

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

    if -8.213216247196925e+129 < b < 6.088267304256603e-81

    1. Initial program 12.3

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

    2. Simplified12.4

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

    4. Applied clear-num12.5

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

    if 6.088267304256603e-81 < b

    1. Initial program 52.2

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

    2. Simplified52.2

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

    3. Taylor expanded around inf 10.5

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.213216247196925388401125773743990732555 \cdot 10^{129}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{a}, -2, 2 \cdot \frac{c}{b}\right)}{2}\\ \mathbf{elif}\;b \le 6.088267304256603437292930310963869002155 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)} - b}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :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)))