Average Error: 34.0 → 7.2
Time: 4.8s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.6114837319571935 \cdot 10^{38}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -3.49799969143154611 \cdot 10^{-295}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le 1.5018928133646766 \cdot 10^{85}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -6.6114837319571935 \cdot 10^{38}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -3.49799969143154611 \cdot 10^{-295}:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\

\mathbf{elif}\;b \le 1.5018928133646766 \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 \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r84720 = b;
        double r84721 = -r84720;
        double r84722 = r84720 * r84720;
        double r84723 = 4.0;
        double r84724 = a;
        double r84725 = c;
        double r84726 = r84724 * r84725;
        double r84727 = r84723 * r84726;
        double r84728 = r84722 - r84727;
        double r84729 = sqrt(r84728);
        double r84730 = r84721 - r84729;
        double r84731 = 2.0;
        double r84732 = r84731 * r84724;
        double r84733 = r84730 / r84732;
        return r84733;
}
↓
double f(double a, double b, double c) {
        double r84734 = b;
        double r84735 = -6.6114837319571935e+38;
        bool r84736 = r84734 <= r84735;
        double r84737 = -1.0;
        double r84738 = c;
        double r84739 = r84738 / r84734;
        double r84740 = r84737 * r84739;
        double r84741 = -3.497999691431546e-295;
        bool r84742 = r84734 <= r84741;
        double r84743 = 2.0;
        double r84744 = r84743 * r84738;
        double r84745 = -r84734;
        double r84746 = r84734 * r84734;
        double r84747 = 4.0;
        double r84748 = a;
        double r84749 = r84748 * r84738;
        double r84750 = r84747 * r84749;
        double r84751 = r84746 - r84750;
        double r84752 = sqrt(r84751);
        double r84753 = r84745 + r84752;
        double r84754 = r84744 / r84753;
        double r84755 = 1.5018928133646766e+85;
        bool r84756 = r84734 <= r84755;
        double r84757 = r84743 * r84748;
        double r84758 = r84745 / r84757;
        double r84759 = r84752 / r84757;
        double r84760 = r84758 - r84759;
        double r84761 = 1.0;
        double r84762 = r84734 / r84748;
        double r84763 = r84739 - r84762;
        double r84764 = r84761 * r84763;
        double r84765 = r84756 ? r84760 : r84764;
        double r84766 = r84742 ? r84754 : r84765;
        double r84767 = r84736 ? r84740 : r84766;
        return r84767;
}

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
Herbie7.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -6.6114837319571935e+38

    1. Initial program 57.0

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

    2. Taylor expanded around -inf 4.8

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

    if -6.6114837319571935e+38 < b < -3.497999691431546e-295

    1. Initial program 29.3

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

    3. Applied div-inv29.4

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

    5. Applied flip--29.4

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

    6. Applied associate-*l/29.4

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

    7. Simplified17.6

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

    8. Taylor expanded around 0 10.1

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

    if -3.497999691431546e-295 < b < 1.5018928133646766e+85

    1. Initial program 9.4

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

    3. Applied div-sub9.4

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

    if 1.5018928133646766e+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. Taylor expanded around inf 3.6

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

    3. Simplified3.6

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification7.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.6114837319571935 \cdot 10^{38}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -3.49799969143154611 \cdot 10^{-295}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le 1.5018928133646766 \cdot 10^{85}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(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)))