Average Error: 34.5 → 9.9
Time: 9.4s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -16283081095555591571721024035618816:\\ \;\;\;\;1 \cdot \left(-1 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \le \frac{-868252521030753}{3.634193621478034452746619039440022671768 \cdot 10^{134}}:\\ \;\;\;\;\frac{1}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b} \cdot \frac{\frac{4 \cdot \left(a \cdot c\right)}{2}}{a}\\ \mathbf{elif}\;b \le \frac{-8402861763014741}{4.994797680505587570210555567669066089198 \cdot 10^{145}}:\\ \;\;\;\;1 \cdot \left(-1 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \le \frac{3369331144082303}{11150372599265311570767859136324180752990000}:\\ \;\;\;\;1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -16283081095555591571721024035618816:\\
\;\;\;\;1 \cdot \left(-1 \cdot \frac{c}{b}\right)\\

\mathbf{elif}\;b \le \frac{-868252521030753}{3.634193621478034452746619039440022671768 \cdot 10^{134}}:\\
\;\;\;\;\frac{1}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b} \cdot \frac{\frac{4 \cdot \left(a \cdot c\right)}{2}}{a}\\

\mathbf{elif}\;b \le \frac{-8402861763014741}{4.994797680505587570210555567669066089198 \cdot 10^{145}}:\\
\;\;\;\;1 \cdot \left(-1 \cdot \frac{c}{b}\right)\\

\mathbf{elif}\;b \le \frac{3369331144082303}{11150372599265311570767859136324180752990000}:\\
\;\;\;\;1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r288852 = b;
        double r288853 = -r288852;
        double r288854 = r288852 * r288852;
        double r288855 = 4.0;
        double r288856 = a;
        double r288857 = c;
        double r288858 = r288856 * r288857;
        double r288859 = r288855 * r288858;
        double r288860 = r288854 - r288859;
        double r288861 = sqrt(r288860);
        double r288862 = r288853 - r288861;
        double r288863 = 2.0;
        double r288864 = r288863 * r288856;
        double r288865 = r288862 / r288864;
        return r288865;
}
↓
double f(double a, double b, double c) {
        double r288866 = b;
        double r288867 = -1.6283081095555592e+34;
        bool r288868 = r288866 <= r288867;
        double r288869 = 1.0;
        double r288870 = -1.0;
        double r288871 = c;
        double r288872 = r288871 / r288866;
        double r288873 = r288870 * r288872;
        double r288874 = r288869 * r288873;
        double r288875 = -868252521030753.0;
        double r288876 = 3.6341936214780345e+134;
        double r288877 = r288875 / r288876;
        bool r288878 = r288866 <= r288877;
        double r288879 = r288866 * r288866;
        double r288880 = 4.0;
        double r288881 = a;
        double r288882 = r288881 * r288871;
        double r288883 = r288880 * r288882;
        double r288884 = r288879 - r288883;
        double r288885 = sqrt(r288884);
        double r288886 = r288885 - r288866;
        double r288887 = r288869 / r288886;
        double r288888 = 2.0;
        double r288889 = r288883 / r288888;
        double r288890 = r288889 / r288881;
        double r288891 = r288887 * r288890;
        double r288892 = -8402861763014741.0;
        double r288893 = 4.994797680505588e+145;
        double r288894 = r288892 / r288893;
        bool r288895 = r288866 <= r288894;
        double r288896 = 3369331144082303.0;
        double r288897 = 1.1150372599265312e+43;
        double r288898 = r288896 / r288897;
        bool r288899 = r288866 <= r288898;
        double r288900 = -r288866;
        double r288901 = r288900 - r288885;
        double r288902 = r288888 * r288881;
        double r288903 = r288901 / r288902;
        double r288904 = r288869 * r288903;
        double r288905 = 1.0;
        double r288906 = r288866 / r288881;
        double r288907 = r288872 - r288906;
        double r288908 = r288905 * r288907;
        double r288909 = r288899 ? r288904 : r288908;
        double r288910 = r288895 ? r288874 : r288909;
        double r288911 = r288878 ? r288891 : r288910;
        double r288912 = r288868 ? r288874 : r288911;
        return r288912;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original34.5
Target20.9
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.6283081095555592e+34 or -2.3891201500640816e-120 < b < -1.682322748689228e-130

    1. Initial program 55.5

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

    3. Applied div-inv55.5

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

    5. Applied *-un-lft-identity55.5

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

    6. Applied associate-*l*55.5

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

    7. Simplified55.5

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

    8. Taylor expanded around -inf 5.6

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

    if -1.6283081095555592e+34 < b < -2.3891201500640816e-120

    1. Initial program 39.8

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

    3. Applied div-inv39.8

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

    5. Applied flip--39.9

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

    6. Applied frac-times43.6

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

    7. Simplified22.7

      \[\leadsto \frac{\color{blue}{0 + 1 \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \left(2 \cdot a\right)}\]
    8. Using strategy rm

    9. Applied *-un-lft-identity22.7

      \[\leadsto \frac{\color{blue}{1 \cdot \left(0 + 1 \cdot \left(4 \cdot \left(a \cdot c\right)\right)\right)}}{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \left(2 \cdot a\right)}\]

    10. Applied times-frac17.3

      \[\leadsto \color{blue}{\frac{1}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \frac{0 + 1 \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{2 \cdot a}}\]

    11. Simplified17.3

      \[\leadsto \color{blue}{\frac{1}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}} \cdot \frac{0 + 1 \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{2 \cdot a}\]

    12. Simplified17.3

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

    if -1.682322748689228e-130 < b < 3.0217206771227585e-28

    1. Initial program 12.8

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

    3. Applied div-inv12.9

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

    5. Applied *-un-lft-identity12.9

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

    6. Applied associate-*l*12.9

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

    7. Simplified12.8

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

    if 3.0217206771227585e-28 < b

    1. Initial program 31.0

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

    2. Taylor expanded around inf 8.0

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

    3. Simplified8.0

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -16283081095555591571721024035618816:\\ \;\;\;\;1 \cdot \left(-1 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \le \frac{-868252521030753}{3.634193621478034452746619039440022671768 \cdot 10^{134}}:\\ \;\;\;\;\frac{1}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b} \cdot \frac{\frac{4 \cdot \left(a \cdot c\right)}{2}}{a}\\ \mathbf{elif}\;b \le \frac{-8402861763014741}{4.994797680505587570210555567669066089198 \cdot 10^{145}}:\\ \;\;\;\;1 \cdot \left(-1 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \le \frac{3369331144082303}{11150372599265311570767859136324180752990000}:\\ \;\;\;\;1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 350497007 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ 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)))