Average Error: 33.9 → 8.8
Time: 5.4s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.860355455153336511492756422997945623034 \cdot 10^{55}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -3.525283965541116465664395093099544846334 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le 2.368471338029900067384853691743532212305 \cdot 10^{94}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -3.860355455153336511492756422997945623034 \cdot 10^{55}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -3.525283965541116465664395093099544846334 \cdot 10^{-144}:\\
\;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\

\mathbf{elif}\;b \le 2.368471338029900067384853691743532212305 \cdot 10^{94}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{\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 r49891 = b;
        double r49892 = -r49891;
        double r49893 = r49891 * r49891;
        double r49894 = 4.0;
        double r49895 = a;
        double r49896 = c;
        double r49897 = r49895 * r49896;
        double r49898 = r49894 * r49897;
        double r49899 = r49893 - r49898;
        double r49900 = sqrt(r49899);
        double r49901 = r49892 - r49900;
        double r49902 = 2.0;
        double r49903 = r49902 * r49895;
        double r49904 = r49901 / r49903;
        return r49904;
}
↓
double f(double a, double b, double c) {
        double r49905 = b;
        double r49906 = -3.8603554551533365e+55;
        bool r49907 = r49905 <= r49906;
        double r49908 = -1.0;
        double r49909 = c;
        double r49910 = r49909 / r49905;
        double r49911 = r49908 * r49910;
        double r49912 = -3.5252839655411165e-144;
        bool r49913 = r49905 <= r49912;
        double r49914 = 4.0;
        double r49915 = a;
        double r49916 = r49915 * r49909;
        double r49917 = r49914 * r49916;
        double r49918 = r49905 * r49905;
        double r49919 = r49918 - r49917;
        double r49920 = sqrt(r49919);
        double r49921 = r49920 - r49905;
        double r49922 = r49917 / r49921;
        double r49923 = 2.0;
        double r49924 = r49923 * r49915;
        double r49925 = r49922 / r49924;
        double r49926 = 2.3684713380299e+94;
        bool r49927 = r49905 <= r49926;
        double r49928 = -r49905;
        double r49929 = r49928 / r49924;
        double r49930 = r49920 / r49924;
        double r49931 = r49929 - r49930;
        double r49932 = 1.0;
        double r49933 = r49905 / r49915;
        double r49934 = r49910 - r49933;
        double r49935 = r49932 * r49934;
        double r49936 = r49927 ? r49931 : r49935;
        double r49937 = r49913 ? r49925 : r49936;
        double r49938 = r49907 ? r49911 : r49937;
        return r49938;
}

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

Original33.9
Target21.0
Herbie8.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.8603554551533365e+55

    1. Initial program 57.7

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

    2. Taylor expanded around -inf 3.8

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

    if -3.8603554551533365e+55 < b < -3.5252839655411165e-144

    1. Initial program 36.9

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

    3. Applied add-sqr-sqrt36.9

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

    4. Applied sqrt-prod37.7

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

    6. Applied flip--37.7

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

    7. Simplified17.4

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

    8. Simplified17.3

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

    if -3.5252839655411165e-144 < b < 2.3684713380299e+94

    1. Initial program 11.1

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

    3. Applied div-sub11.1

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

    if 2.3684713380299e+94 < b

    1. Initial program 44.8

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

    2. Taylor expanded around inf 3.8

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

    3. Simplified3.8

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

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.860355455153336511492756422997945623034 \cdot 10^{55}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -3.525283965541116465664395093099544846334 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le 2.368471338029900067384853691743532212305 \cdot 10^{94}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019346 
(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)))