Average Error: 33.0 → 6.6
Time: 59.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.139254247068609 \cdot 10^{+140}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 9.931313556234952 \cdot 10^{-296}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a} \cdot \frac{1}{2}\\ \mathbf{elif}\;b \le 6.523631550102089 \cdot 10^{+89}:\\ \;\;\;\;\frac{c \cdot -2}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -9.139254247068609 \cdot 10^{+140}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 9.931313556234952 \cdot 10^{-296}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a} \cdot \frac{1}{2}\\

\mathbf{elif}\;b \le 6.523631550102089 \cdot 10^{+89}:\\
\;\;\;\;\frac{c \cdot -2}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}\\

\mathbf{else}:\\
\;\;\;\;-\frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r4047994 = b;
        double r4047995 = -r4047994;
        double r4047996 = r4047994 * r4047994;
        double r4047997 = 4.0;
        double r4047998 = a;
        double r4047999 = r4047997 * r4047998;
        double r4048000 = c;
        double r4048001 = r4047999 * r4048000;
        double r4048002 = r4047996 - r4048001;
        double r4048003 = sqrt(r4048002);
        double r4048004 = r4047995 + r4048003;
        double r4048005 = 2.0;
        double r4048006 = r4048005 * r4047998;
        double r4048007 = r4048004 / r4048006;
        return r4048007;
}
↓
double f(double a, double b, double c) {
        double r4048008 = b;
        double r4048009 = -9.139254247068609e+140;
        bool r4048010 = r4048008 <= r4048009;
        double r4048011 = c;
        double r4048012 = r4048011 / r4048008;
        double r4048013 = a;
        double r4048014 = r4048008 / r4048013;
        double r4048015 = r4048012 - r4048014;
        double r4048016 = 9.931313556234952e-296;
        bool r4048017 = r4048008 <= r4048016;
        double r4048018 = r4048008 * r4048008;
        double r4048019 = r4048011 * r4048013;
        double r4048020 = 4.0;
        double r4048021 = r4048019 * r4048020;
        double r4048022 = r4048018 - r4048021;
        double r4048023 = sqrt(r4048022);
        double r4048024 = r4048023 - r4048008;
        double r4048025 = r4048024 / r4048013;
        double r4048026 = 0.5;
        double r4048027 = r4048025 * r4048026;
        double r4048028 = 6.523631550102089e+89;
        bool r4048029 = r4048008 <= r4048028;
        double r4048030 = -2.0;
        double r4048031 = r4048011 * r4048030;
        double r4048032 = r4048023 + r4048008;
        double r4048033 = r4048031 / r4048032;
        double r4048034 = -r4048012;
        double r4048035 = r4048029 ? r4048033 : r4048034;
        double r4048036 = r4048017 ? r4048027 : r4048035;
        double r4048037 = r4048010 ? r4048015 : r4048036;
        return r4048037;
}

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

Derivation

  1. Split input into 4 regimes

  2. if b < -9.139254247068609e+140

    1. Initial program 55.8

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

    2. Simplified55.8

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

    3. Taylor expanded around -inf 1.6

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

    if -9.139254247068609e+140 < b < 9.931313556234952e-296

    1. Initial program 8.7

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

    2. Simplified8.7

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

    4. Applied *-un-lft-identity8.7

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

    5. Applied associate-/l*8.9

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

    7. Applied *-un-lft-identity8.9

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

    8. Applied *-un-lft-identity8.9

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

    9. Applied distribute-lft-out--8.9

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

    10. Applied times-frac8.8

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

    11. Applied add-sqr-sqrt8.8

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

    12. Applied times-frac8.8

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

    13. Simplified8.8

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

    14. Simplified8.7

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

    if 9.931313556234952e-296 < b < 6.523631550102089e+89

    1. Initial program 32.6

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

    2. Simplified32.6

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

    4. Applied *-un-lft-identity32.6

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

    5. Applied associate-/l*32.6

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

    7. Applied flip--32.7

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

    8. Applied associate-/r/32.8

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

    9. Applied *-un-lft-identity32.8

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

    10. Applied times-frac32.8

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

    11. Simplified16.4

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

    12. Taylor expanded around inf 9.3

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

    14. Applied un-div-inv9.2

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

    if 6.523631550102089e+89 < b

    1. Initial program 58.7

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

    2. Simplified58.7

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

    3. Taylor expanded around inf 2.6

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

    4. Simplified2.6

      \[\leadsto \color{blue}{-\frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.139254247068609 \cdot 10^{+140}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 9.931313556234952 \cdot 10^{-296}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a} \cdot \frac{1}{2}\\ \mathbf{elif}\;b \le 6.523631550102089 \cdot 10^{+89}:\\ \;\;\;\;\frac{c \cdot -2}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))