Average Error: 34.1 → 10.5
Time: 20.3s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.332433396832084322962138528577137922234 \cdot 10^{-58}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 4.825478720088060668779950456669858064189 \cdot 10^{107}:\\ \;\;\;\;\left(-\frac{b}{2 \cdot a}\right) - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]
\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 -9.332433396832084322962138528577137922234 \cdot 10^{-58}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 4.825478720088060668779950456669858064189 \cdot 10^{107}:\\
\;\;\;\;\left(-\frac{b}{2 \cdot a}\right) - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{b}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r4788063 = b;
        double r4788064 = -r4788063;
        double r4788065 = r4788063 * r4788063;
        double r4788066 = 4.0;
        double r4788067 = a;
        double r4788068 = c;
        double r4788069 = r4788067 * r4788068;
        double r4788070 = r4788066 * r4788069;
        double r4788071 = r4788065 - r4788070;
        double r4788072 = sqrt(r4788071);
        double r4788073 = r4788064 - r4788072;
        double r4788074 = 2.0;
        double r4788075 = r4788074 * r4788067;
        double r4788076 = r4788073 / r4788075;
        return r4788076;
}
↓
double f(double a, double b, double c) {
        double r4788077 = b;
        double r4788078 = -9.332433396832084e-58;
        bool r4788079 = r4788077 <= r4788078;
        double r4788080 = -1.0;
        double r4788081 = c;
        double r4788082 = r4788081 / r4788077;
        double r4788083 = r4788080 * r4788082;
        double r4788084 = 4.8254787200880607e+107;
        bool r4788085 = r4788077 <= r4788084;
        double r4788086 = 2.0;
        double r4788087 = a;
        double r4788088 = r4788086 * r4788087;
        double r4788089 = r4788077 / r4788088;
        double r4788090 = -r4788089;
        double r4788091 = r4788077 * r4788077;
        double r4788092 = 4.0;
        double r4788093 = r4788087 * r4788081;
        double r4788094 = r4788092 * r4788093;
        double r4788095 = r4788091 - r4788094;
        double r4788096 = sqrt(r4788095);
        double r4788097 = r4788096 / r4788088;
        double r4788098 = r4788090 - r4788097;
        double r4788099 = r4788077 / r4788087;
        double r4788100 = r4788080 * r4788099;
        double r4788101 = r4788085 ? r4788098 : r4788100;
        double r4788102 = r4788079 ? r4788083 : r4788101;
        return r4788102;
}

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

Original34.1
Target21.4
Herbie10.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -9.332433396832084e-58

    1. Initial program 53.5

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

    2. Taylor expanded around -inf 8.7

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

    if -9.332433396832084e-58 < b < 4.8254787200880607e+107

    1. Initial program 14.1

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

    3. Applied div-sub14.1

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

    if 4.8254787200880607e+107 < b

    1. Initial program 49.2

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

    3. Applied flip--63.2

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

    4. Simplified62.3

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

    5. Simplified62.3

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

    6. Taylor expanded around 0 3.7

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.332433396832084322962138528577137922234 \cdot 10^{-58}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 4.825478720088060668779950456669858064189 \cdot 10^{107}:\\ \;\;\;\;\left(-\frac{b}{2 \cdot a}\right) - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :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)))