Average Error: 34.5 → 10.1
Time: 10.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.28391183709975701 \cdot 10^{106}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.16764411094466422 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}{\frac{2 \cdot a}{\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}}\\ \mathbf{elif}\;b \le -5.52775192595066085 \cdot 10^{-141}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.59909590090097915 \cdot 10^{53}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \end{array}\]
\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 -4.28391183709975701 \cdot 10^{106}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -1.16764411094466422 \cdot 10^{-83}:\\
\;\;\;\;\frac{\frac{1}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}{\frac{2 \cdot a}{\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}}\\

\mathbf{elif}\;b \le -5.52775192595066085 \cdot 10^{-141}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 3.59909590090097915 \cdot 10^{53}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r97900 = b;
        double r97901 = -r97900;
        double r97902 = r97900 * r97900;
        double r97903 = 4.0;
        double r97904 = a;
        double r97905 = c;
        double r97906 = r97904 * r97905;
        double r97907 = r97903 * r97906;
        double r97908 = r97902 - r97907;
        double r97909 = sqrt(r97908);
        double r97910 = r97901 - r97909;
        double r97911 = 2.0;
        double r97912 = r97911 * r97904;
        double r97913 = r97910 / r97912;
        return r97913;
}
↓
double f(double a, double b, double c) {
        double r97914 = b;
        double r97915 = -4.283911837099757e+106;
        bool r97916 = r97914 <= r97915;
        double r97917 = -1.0;
        double r97918 = c;
        double r97919 = r97918 / r97914;
        double r97920 = r97917 * r97919;
        double r97921 = -1.1676441109446642e-83;
        bool r97922 = r97914 <= r97921;
        double r97923 = 1.0;
        double r97924 = r97914 * r97914;
        double r97925 = 4.0;
        double r97926 = a;
        double r97927 = r97926 * r97918;
        double r97928 = r97925 * r97927;
        double r97929 = r97924 - r97928;
        double r97930 = sqrt(r97929);
        double r97931 = r97930 - r97914;
        double r97932 = sqrt(r97931);
        double r97933 = r97923 / r97932;
        double r97934 = 2.0;
        double r97935 = r97934 * r97926;
        double r97936 = r97928 / r97932;
        double r97937 = r97935 / r97936;
        double r97938 = r97933 / r97937;
        double r97939 = -5.527751925950661e-141;
        bool r97940 = r97914 <= r97939;
        double r97941 = 3.599095900900979e+53;
        bool r97942 = r97914 <= r97941;
        double r97943 = -r97914;
        double r97944 = r97943 - r97930;
        double r97945 = r97944 / r97935;
        double r97946 = -2.0;
        double r97947 = r97946 * r97914;
        double r97948 = r97947 / r97935;
        double r97949 = r97942 ? r97945 : r97948;
        double r97950 = r97940 ? r97920 : r97949;
        double r97951 = r97922 ? r97938 : r97950;
        double r97952 = r97916 ? r97920 : r97951;
        return r97952;
}

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
Target20.8
Herbie10.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -4.283911837099757e+106 or -1.1676441109446642e-83 < b < -5.527751925950661e-141

    1. Initial program 54.5

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

    2. Taylor expanded around -inf 8.4

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

    if -4.283911837099757e+106 < b < -1.1676441109446642e-83

    1. Initial program 42.9

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

    3. Applied flip--42.9

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

    4. Simplified15.2

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

    5. Simplified15.2

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

    7. Applied add-sqr-sqrt15.3

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

    8. Applied *-un-lft-identity15.3

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

    9. Applied times-frac15.4

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

    10. Applied associate-/l*14.5

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

    11. Simplified14.5

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

    if -5.527751925950661e-141 < b < 3.599095900900979e+53

    1. Initial program 12.0

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

    if 3.599095900900979e+53 < b

    1. Initial program 38.5

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

    3. Applied flip--62.0

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

    4. Simplified61.3

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

    5. Simplified61.3

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

    6. Taylor expanded around 0 5.4

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{2 \cdot a}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.28391183709975701 \cdot 10^{106}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.16764411094466422 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}{\frac{2 \cdot a}{\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}}\\ \mathbf{elif}\;b \le -5.52775192595066085 \cdot 10^{-141}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.59909590090097915 \cdot 10^{53}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(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)))