Average Error: 34.0 → 11.7
Time: 5.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.92145859080318459 \cdot 10^{30}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.35303540148363475 \cdot 10^{-114}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{elif}\;b \le -3.70403707285546 \cdot 10^{-127}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.3866645776898725 \cdot 10^{85}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]
\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 -5.92145859080318459 \cdot 10^{30}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -1.35303540148363475 \cdot 10^{-114}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

\mathbf{elif}\;b \le -3.70403707285546 \cdot 10^{-127}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 2.3866645776898725 \cdot 10^{85}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \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 r87863 = b;
        double r87864 = -r87863;
        double r87865 = r87863 * r87863;
        double r87866 = 4.0;
        double r87867 = a;
        double r87868 = c;
        double r87869 = r87867 * r87868;
        double r87870 = r87866 * r87869;
        double r87871 = r87865 - r87870;
        double r87872 = sqrt(r87871);
        double r87873 = r87864 - r87872;
        double r87874 = 2.0;
        double r87875 = r87874 * r87867;
        double r87876 = r87873 / r87875;
        return r87876;
}
↓
double f(double a, double b, double c) {
        double r87877 = b;
        double r87878 = -5.921458590803185e+30;
        bool r87879 = r87877 <= r87878;
        double r87880 = -1.0;
        double r87881 = c;
        double r87882 = r87881 / r87877;
        double r87883 = r87880 * r87882;
        double r87884 = -1.3530354014836348e-114;
        bool r87885 = r87877 <= r87884;
        double r87886 = -r87877;
        double r87887 = 2.0;
        double r87888 = a;
        double r87889 = r87887 * r87888;
        double r87890 = r87886 / r87889;
        double r87891 = r87877 * r87877;
        double r87892 = 4.0;
        double r87893 = r87888 * r87881;
        double r87894 = r87892 * r87893;
        double r87895 = r87891 - r87894;
        double r87896 = sqrt(r87895);
        double r87897 = r87896 / r87889;
        double r87898 = r87890 - r87897;
        double r87899 = -3.704037072855455e-127;
        bool r87900 = r87877 <= r87899;
        double r87901 = 2.3866645776898725e+85;
        bool r87902 = r87877 <= r87901;
        double r87903 = r87877 / r87888;
        double r87904 = r87880 * r87903;
        double r87905 = r87902 ? r87898 : r87904;
        double r87906 = r87900 ? r87883 : r87905;
        double r87907 = r87885 ? r87898 : r87906;
        double r87908 = r87879 ? r87883 : r87907;
        return r87908;
}

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

Original34.0
Target21.2
Herbie11.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -5.921458590803185e+30 or -1.3530354014836348e-114 < b < -3.704037072855455e-127

    1. Initial program 56.0

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

    2. Taylor expanded around -inf 6.0

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

    if -5.921458590803185e+30 < b < -1.3530354014836348e-114 or -3.704037072855455e-127 < b < 2.3866645776898725e+85

    1. Initial program 17.6

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

    3. Applied div-sub17.6

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

    if 2.3866645776898725e+85 < b

    1. Initial program 43.7

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

    3. Applied clear-num43.8

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

    4. Taylor expanded around 0 3.8

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

  4. Final simplification11.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.92145859080318459 \cdot 10^{30}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.35303540148363475 \cdot 10^{-114}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{elif}\;b \le -3.70403707285546 \cdot 10^{-127}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.3866645776898725 \cdot 10^{85}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(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)))