Average Error: 34.4 → 10.1
Time: 17.3s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.635925081143504476780080161813975782827 \cdot 10^{-66}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.206904744652339671334892722279467095293 \cdot 10^{101}:\\ \;\;\;\;\frac{-\left(\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} + b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]
\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 -8.635925081143504476780080161813975782827 \cdot 10^{-66}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 3.206904744652339671334892722279467095293 \cdot 10^{101}:\\
\;\;\;\;\frac{-\left(\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} + b\right)}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2819894 = b;
        double r2819895 = -r2819894;
        double r2819896 = r2819894 * r2819894;
        double r2819897 = 4.0;
        double r2819898 = a;
        double r2819899 = c;
        double r2819900 = r2819898 * r2819899;
        double r2819901 = r2819897 * r2819900;
        double r2819902 = r2819896 - r2819901;
        double r2819903 = sqrt(r2819902);
        double r2819904 = r2819895 - r2819903;
        double r2819905 = 2.0;
        double r2819906 = r2819905 * r2819898;
        double r2819907 = r2819904 / r2819906;
        return r2819907;
}
↓
double f(double a, double b, double c) {
        double r2819908 = b;
        double r2819909 = -8.635925081143504e-66;
        bool r2819910 = r2819908 <= r2819909;
        double r2819911 = -1.0;
        double r2819912 = c;
        double r2819913 = r2819912 / r2819908;
        double r2819914 = r2819911 * r2819913;
        double r2819915 = 3.2069047446523397e+101;
        bool r2819916 = r2819908 <= r2819915;
        double r2819917 = r2819908 * r2819908;
        double r2819918 = 4.0;
        double r2819919 = r2819912 * r2819918;
        double r2819920 = a;
        double r2819921 = r2819919 * r2819920;
        double r2819922 = r2819917 - r2819921;
        double r2819923 = sqrt(r2819922);
        double r2819924 = r2819923 + r2819908;
        double r2819925 = -r2819924;
        double r2819926 = 2.0;
        double r2819927 = r2819926 * r2819920;
        double r2819928 = r2819925 / r2819927;
        double r2819929 = r2819908 / r2819920;
        double r2819930 = r2819913 - r2819929;
        double r2819931 = 1.0;
        double r2819932 = r2819930 * r2819931;
        double r2819933 = r2819916 ? r2819928 : r2819932;
        double r2819934 = r2819910 ? r2819914 : r2819933;
        return r2819934;
}

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

Original34.4
Target20.9
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 3 regimes

  2. if b < -8.635925081143504e-66

    1. Initial program 53.4

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

    2. Taylor expanded around -inf 8.4

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

    if -8.635925081143504e-66 < b < 3.2069047446523397e+101

    1. Initial program 13.4

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

    3. Applied div-inv13.5

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

    5. Applied associate-*r/13.4

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

    6. Simplified13.5

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

    if 3.2069047446523397e+101 < b

    1. Initial program 46.8

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

    3. Applied div-inv46.8

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

    5. Applied associate-*r/46.8

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

    6. Simplified46.8

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

    7. Taylor expanded around inf 4.4

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

    8. Simplified4.4

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.635925081143504476780080161813975782827 \cdot 10^{-66}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.206904744652339671334892722279467095293 \cdot 10^{101}:\\ \;\;\;\;\frac{-\left(\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} + b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]

Reproduce

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

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

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