Average Error: 34.5 → 10.5
Time: 10.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.78285893492843261 \cdot 10^{-126}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.6627135292415903 \cdot 10^{111}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]
\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 -4.78285893492843261 \cdot 10^{-126}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 3.6627135292415903 \cdot 10^{111}:\\
\;\;\;\;\frac{\left(-b\right) - \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 r82855 = b;
        double r82856 = -r82855;
        double r82857 = r82855 * r82855;
        double r82858 = 4.0;
        double r82859 = a;
        double r82860 = c;
        double r82861 = r82859 * r82860;
        double r82862 = r82858 * r82861;
        double r82863 = r82857 - r82862;
        double r82864 = sqrt(r82863);
        double r82865 = r82856 - r82864;
        double r82866 = 2.0;
        double r82867 = r82866 * r82859;
        double r82868 = r82865 / r82867;
        return r82868;
}
↓
double f(double a, double b, double c) {
        double r82869 = b;
        double r82870 = -4.7828589349284326e-126;
        bool r82871 = r82869 <= r82870;
        double r82872 = -1.0;
        double r82873 = c;
        double r82874 = r82873 / r82869;
        double r82875 = r82872 * r82874;
        double r82876 = 3.6627135292415903e+111;
        bool r82877 = r82869 <= r82876;
        double r82878 = -r82869;
        double r82879 = r82869 * r82869;
        double r82880 = 4.0;
        double r82881 = a;
        double r82882 = r82881 * r82873;
        double r82883 = r82880 * r82882;
        double r82884 = r82879 - r82883;
        double r82885 = sqrt(r82884);
        double r82886 = r82878 - r82885;
        double r82887 = 2.0;
        double r82888 = r82887 * r82881;
        double r82889 = r82886 / r82888;
        double r82890 = r82869 / r82881;
        double r82891 = r82872 * r82890;
        double r82892 = r82877 ? r82889 : r82891;
        double r82893 = r82871 ? r82875 : r82892;
        return r82893;
}

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

Original34.5
Target20.6
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 < -4.7828589349284326e-126

    1. Initial program 51.3

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

    2. Taylor expanded around -inf 11.3

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

    if -4.7828589349284326e-126 < b < 3.6627135292415903e+111

    1. Initial program 12.0

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

    if 3.6627135292415903e+111 < b

    1. Initial program 49.7

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

    3. Applied flip--63.3

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

    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 3.6

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.78285893492843261 \cdot 10^{-126}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.6627135292415903 \cdot 10^{111}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :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)))