Average Error: 33.7 → 10.7
Time: 17.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.363255598823911 \cdot 10^{-15}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -6.936587154412951 \cdot 10^{-28}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{2 \cdot a} \cdot \sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c}\\ \mathbf{elif}\;b \le -2.3344326820285623 \cdot 10^{-123}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.6691257204922504 \cdot 10^{+85}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{2 \cdot a} \cdot \sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -7.363255598823911 \cdot 10^{-15}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le -6.936587154412951 \cdot 10^{-28}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{2 \cdot a} \cdot \sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c}\\

\mathbf{elif}\;b \le -2.3344326820285623 \cdot 10^{-123}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 1.6691257204922504 \cdot 10^{+85}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{2 \cdot a} \cdot \sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r3655959 = b;
        double r3655960 = -r3655959;
        double r3655961 = r3655959 * r3655959;
        double r3655962 = 4.0;
        double r3655963 = a;
        double r3655964 = c;
        double r3655965 = r3655963 * r3655964;
        double r3655966 = r3655962 * r3655965;
        double r3655967 = r3655961 - r3655966;
        double r3655968 = sqrt(r3655967);
        double r3655969 = r3655960 - r3655968;
        double r3655970 = 2.0;
        double r3655971 = r3655970 * r3655963;
        double r3655972 = r3655969 / r3655971;
        return r3655972;
}
↓
double f(double a, double b, double c) {
        double r3655973 = b;
        double r3655974 = -7.363255598823911e-15;
        bool r3655975 = r3655973 <= r3655974;
        double r3655976 = c;
        double r3655977 = r3655976 / r3655973;
        double r3655978 = -r3655977;
        double r3655979 = -6.936587154412951e-28;
        bool r3655980 = r3655973 <= r3655979;
        double r3655981 = -r3655973;
        double r3655982 = 2.0;
        double r3655983 = a;
        double r3655984 = r3655982 * r3655983;
        double r3655985 = r3655981 / r3655984;
        double r3655986 = 1.0;
        double r3655987 = r3655986 / r3655984;
        double r3655988 = r3655973 * r3655973;
        double r3655989 = -4.0;
        double r3655990 = r3655989 * r3655983;
        double r3655991 = r3655990 * r3655976;
        double r3655992 = r3655988 + r3655991;
        double r3655993 = sqrt(r3655992);
        double r3655994 = r3655987 * r3655993;
        double r3655995 = r3655985 - r3655994;
        double r3655996 = -2.3344326820285623e-123;
        bool r3655997 = r3655973 <= r3655996;
        double r3655998 = 1.6691257204922504e+85;
        bool r3655999 = r3655973 <= r3655998;
        double r3656000 = r3655973 / r3655983;
        double r3656001 = r3655977 - r3656000;
        double r3656002 = r3655999 ? r3655995 : r3656001;
        double r3656003 = r3655997 ? r3655978 : r3656002;
        double r3656004 = r3655980 ? r3655995 : r3656003;
        double r3656005 = r3655975 ? r3655978 : r3656004;
        return r3656005;
}

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

Original33.7
Target21.0
Herbie10.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -7.363255598823911e-15 or -6.936587154412951e-28 < b < -2.3344326820285623e-123

    1. Initial program 50.9

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

    2. Taylor expanded around -inf 10.6

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

    3. Simplified10.6

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

    if -7.363255598823911e-15 < b < -6.936587154412951e-28 or -2.3344326820285623e-123 < b < 1.6691257204922504e+85

    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 sub-neg13.4

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

    4. Simplified13.4

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

    6. Applied div-sub13.4

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

    8. Applied div-inv13.5

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

    if 1.6691257204922504e+85 < b

    1. Initial program 42.9

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

    2. Taylor expanded around inf 3.7

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

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.363255598823911 \cdot 10^{-15}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -6.936587154412951 \cdot 10^{-28}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{2 \cdot a} \cdot \sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c}\\ \mathbf{elif}\;b \le -2.3344326820285623 \cdot 10^{-123}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.6691257204922504 \cdot 10^{+85}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{2 \cdot a} \cdot \sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (a b c)
  :name "quadm (p42, negative)"

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