Average Error: 34.3 → 10.3
Time: 8.4s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.525509213201143173482122987435994097044 \cdot 10^{-76}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.112435874232008033470805388544595401902 \cdot 10^{74}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -1.525509213201143173482122987435994097044 \cdot 10^{-76}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.112435874232008033470805388544595401902 \cdot 10^{74}:\\
\;\;\;\;\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 r91063 = b;
        double r91064 = -r91063;
        double r91065 = r91063 * r91063;
        double r91066 = 4.0;
        double r91067 = a;
        double r91068 = c;
        double r91069 = r91067 * r91068;
        double r91070 = r91066 * r91069;
        double r91071 = r91065 - r91070;
        double r91072 = sqrt(r91071);
        double r91073 = r91064 - r91072;
        double r91074 = 2.0;
        double r91075 = r91074 * r91067;
        double r91076 = r91073 / r91075;
        return r91076;
}
↓
double f(double a, double b, double c) {
        double r91077 = b;
        double r91078 = -1.5255092132011432e-76;
        bool r91079 = r91077 <= r91078;
        double r91080 = -1.0;
        double r91081 = c;
        double r91082 = r91081 / r91077;
        double r91083 = r91080 * r91082;
        double r91084 = 1.112435874232008e+74;
        bool r91085 = r91077 <= r91084;
        double r91086 = -r91077;
        double r91087 = r91077 * r91077;
        double r91088 = 4.0;
        double r91089 = a;
        double r91090 = r91089 * r91081;
        double r91091 = r91088 * r91090;
        double r91092 = r91087 - r91091;
        double r91093 = sqrt(r91092);
        double r91094 = r91086 - r91093;
        double r91095 = 2.0;
        double r91096 = r91095 * r91089;
        double r91097 = r91094 / r91096;
        double r91098 = 1.0;
        double r91099 = r91077 / r91089;
        double r91100 = r91082 - r91099;
        double r91101 = r91098 * r91100;
        double r91102 = r91085 ? r91097 : r91101;
        double r91103 = r91079 ? r91083 : r91102;
        return r91103;
}

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

Original34.3
Target21.3
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.5255092132011432e-76

    1. Initial program 53.0

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

    2. Taylor expanded around -inf 9.2

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

    if -1.5255092132011432e-76 < b < 1.112435874232008e+74

    1. Initial program 13.5

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

    if 1.112435874232008e+74 < b

    1. Initial program 43.0

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

    2. Taylor expanded around inf 5.0

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

    3. Simplified5.0

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.525509213201143173482122987435994097044 \cdot 10^{-76}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.112435874232008033470805388544595401902 \cdot 10^{74}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019308 
(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)))