Average Error: 34.1 → 10.5
Time: 20.5s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.332433396832084322962138528577137922234 \cdot 10^{-58}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 4.825478720088060668779950456669858064189 \cdot 10^{107}:\\ \;\;\;\;\left(-\frac{b}{2 \cdot a}\right) - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{b}{2 \cdot a}\right) - \frac{b}{a} \cdot 0.5\\ \end{array}\]
\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}:\\
\;\;\;\;\left(-\frac{b}{2 \cdot a}\right) - \frac{b}{a} \cdot 0.5\\

\end{array}
double f(double a, double b, double c) {
        double r3814841 = b;
        double r3814842 = -r3814841;
        double r3814843 = r3814841 * r3814841;
        double r3814844 = 4.0;
        double r3814845 = a;
        double r3814846 = c;
        double r3814847 = r3814845 * r3814846;
        double r3814848 = r3814844 * r3814847;
        double r3814849 = r3814843 - r3814848;
        double r3814850 = sqrt(r3814849);
        double r3814851 = r3814842 - r3814850;
        double r3814852 = 2.0;
        double r3814853 = r3814852 * r3814845;
        double r3814854 = r3814851 / r3814853;
        return r3814854;
}
↓
double f(double a, double b, double c) {
        double r3814855 = b;
        double r3814856 = -9.332433396832084e-58;
        bool r3814857 = r3814855 <= r3814856;
        double r3814858 = -1.0;
        double r3814859 = c;
        double r3814860 = r3814859 / r3814855;
        double r3814861 = r3814858 * r3814860;
        double r3814862 = 4.8254787200880607e+107;
        bool r3814863 = r3814855 <= r3814862;
        double r3814864 = 2.0;
        double r3814865 = a;
        double r3814866 = r3814864 * r3814865;
        double r3814867 = r3814855 / r3814866;
        double r3814868 = -r3814867;
        double r3814869 = r3814855 * r3814855;
        double r3814870 = 4.0;
        double r3814871 = r3814865 * r3814859;
        double r3814872 = r3814870 * r3814871;
        double r3814873 = r3814869 - r3814872;
        double r3814874 = sqrt(r3814873);
        double r3814875 = r3814874 / r3814866;
        double r3814876 = r3814868 - r3814875;
        double r3814877 = r3814855 / r3814865;
        double r3814878 = 0.5;
        double r3814879 = r3814877 * r3814878;
        double r3814880 = r3814868 - r3814879;
        double r3814881 = r3814863 ? r3814876 : r3814880;
        double r3814882 = r3814857 ? r3814861 : r3814881;
        return r3814882;
}

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 div-sub49.2

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

    5. Applied clear-num49.2

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

    6. Taylor expanded around 0 3.7

      \[\leadsto \frac{-b}{2 \cdot a} - \color{blue}{0.5 \cdot \frac{b}{a}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.332433396832084322962138528577137922234 \cdot 10^{-58}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 4.825478720088060668779950456669858064189 \cdot 10^{107}:\\ \;\;\;\;\left(-\frac{b}{2 \cdot a}\right) - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{b}{2 \cdot a}\right) - \frac{b}{a} \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(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)))