Average Error: 34.0 → 9.9
Time: 13.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.683005404599734610349454603271447966358 \cdot 10^{80}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -6.982634002611897496887233296015886335875 \cdot 10^{-268}:\\ \;\;\;\;-\frac{\frac{c \cdot \left(4 \cdot a\right)}{b - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}{2 \cdot a}\\ \mathbf{elif}\;b \le 2098867031.934578418731689453125:\\ \;\;\;\;\frac{-\left(b + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -1.683005404599734610349454603271447966358 \cdot 10^{80}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -6.982634002611897496887233296015886335875 \cdot 10^{-268}:\\
\;\;\;\;-\frac{\frac{c \cdot \left(4 \cdot a\right)}{b - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}{2 \cdot a}\\

\mathbf{elif}\;b \le 2098867031.934578418731689453125:\\
\;\;\;\;\frac{-\left(b + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\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 r107701 = b;
        double r107702 = -r107701;
        double r107703 = r107701 * r107701;
        double r107704 = 4.0;
        double r107705 = a;
        double r107706 = c;
        double r107707 = r107705 * r107706;
        double r107708 = r107704 * r107707;
        double r107709 = r107703 - r107708;
        double r107710 = sqrt(r107709);
        double r107711 = r107702 - r107710;
        double r107712 = 2.0;
        double r107713 = r107712 * r107705;
        double r107714 = r107711 / r107713;
        return r107714;
}
↓
double f(double a, double b, double c) {
        double r107715 = b;
        double r107716 = -1.6830054045997346e+80;
        bool r107717 = r107715 <= r107716;
        double r107718 = -1.0;
        double r107719 = c;
        double r107720 = r107719 / r107715;
        double r107721 = r107718 * r107720;
        double r107722 = -6.982634002611897e-268;
        bool r107723 = r107715 <= r107722;
        double r107724 = 4.0;
        double r107725 = a;
        double r107726 = r107724 * r107725;
        double r107727 = r107719 * r107726;
        double r107728 = r107715 * r107715;
        double r107729 = r107725 * r107719;
        double r107730 = r107729 * r107724;
        double r107731 = r107728 - r107730;
        double r107732 = sqrt(r107731);
        double r107733 = r107715 - r107732;
        double r107734 = r107727 / r107733;
        double r107735 = 2.0;
        double r107736 = r107735 * r107725;
        double r107737 = r107734 / r107736;
        double r107738 = -r107737;
        double r107739 = 2098867031.9345784;
        bool r107740 = r107715 <= r107739;
        double r107741 = r107715 + r107732;
        double r107742 = -r107741;
        double r107743 = r107742 / r107736;
        double r107744 = 1.0;
        double r107745 = r107715 / r107725;
        double r107746 = r107720 - r107745;
        double r107747 = r107744 * r107746;
        double r107748 = r107740 ? r107743 : r107747;
        double r107749 = r107723 ? r107738 : r107748;
        double r107750 = r107717 ? r107721 : r107749;
        return r107750;
}

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
Target20.9
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.6830054045997346e+80

    1. Initial program 58.0

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

    2. Taylor expanded around -inf 2.8

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

    if -1.6830054045997346e+80 < b < -6.982634002611897e-268

    1. Initial program 33.7

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

    3. Applied div-inv33.8

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

    5. Applied pow133.8

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

    6. Applied pow133.8

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

    7. Applied pow-prod-down33.8

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

    8. Simplified33.7

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

    10. Applied flip-+33.8

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

    11. Simplified17.2

      \[\leadsto {\left(\frac{-\frac{\color{blue}{0 + c \cdot \left(4 \cdot a\right)}}{b - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}{2 \cdot a}\right)}^{1}\]

    if -6.982634002611897e-268 < b < 2098867031.9345784

    1. Initial program 11.8

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

    3. Applied div-inv11.9

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

    5. Applied pow111.9

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

    6. Applied pow111.9

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

    7. Applied pow-prod-down11.9

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

    8. Simplified11.8

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

    if 2098867031.9345784 < b

    1. Initial program 32.3

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

    2. Taylor expanded around inf 7.0

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

    3. Simplified7.0

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.683005404599734610349454603271447966358 \cdot 10^{80}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -6.982634002611897496887233296015886335875 \cdot 10^{-268}:\\ \;\;\;\;-\frac{\frac{c \cdot \left(4 \cdot a\right)}{b - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}{2 \cdot a}\\ \mathbf{elif}\;b \le 2098867031.934578418731689453125:\\ \;\;\;\;\frac{-\left(b + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +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)))