Average Error: 33.9 → 7.3
Time: 54.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.157094219357017 \cdot 10^{+135}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 9.088113400659685 \cdot 10^{-185}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 1.8091015183831773 \cdot 10^{+43}:\\ \;\;\;\;\frac{1}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + b} \cdot \left(\sqrt{\frac{1}{2}} \cdot \left(\frac{c}{\frac{-1}{4}} \cdot \sqrt{\frac{1}{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -3.157094219357017 \cdot 10^{+135}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 9.088113400659685 \cdot 10^{-185}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\

\mathbf{elif}\;b \le 1.8091015183831773 \cdot 10^{+43}:\\
\;\;\;\;\frac{1}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + b} \cdot \left(\sqrt{\frac{1}{2}} \cdot \left(\frac{c}{\frac{-1}{4}} \cdot \sqrt{\frac{1}{2}}\right)\right)\\

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

\end{array}
double f(double a, double b, double c) {
        double r7945480 = b;
        double r7945481 = -r7945480;
        double r7945482 = r7945480 * r7945480;
        double r7945483 = 4.0;
        double r7945484 = a;
        double r7945485 = c;
        double r7945486 = r7945484 * r7945485;
        double r7945487 = r7945483 * r7945486;
        double r7945488 = r7945482 - r7945487;
        double r7945489 = sqrt(r7945488);
        double r7945490 = r7945481 + r7945489;
        double r7945491 = 2.0;
        double r7945492 = r7945491 * r7945484;
        double r7945493 = r7945490 / r7945492;
        return r7945493;
}
↓
double f(double a, double b, double c) {
        double r7945494 = b;
        double r7945495 = -3.157094219357017e+135;
        bool r7945496 = r7945494 <= r7945495;
        double r7945497 = c;
        double r7945498 = r7945497 / r7945494;
        double r7945499 = a;
        double r7945500 = r7945494 / r7945499;
        double r7945501 = r7945498 - r7945500;
        double r7945502 = 9.088113400659685e-185;
        bool r7945503 = r7945494 <= r7945502;
        double r7945504 = r7945494 * r7945494;
        double r7945505 = 4.0;
        double r7945506 = r7945499 * r7945505;
        double r7945507 = r7945506 * r7945497;
        double r7945508 = r7945504 - r7945507;
        double r7945509 = sqrt(r7945508);
        double r7945510 = r7945509 - r7945494;
        double r7945511 = 2.0;
        double r7945512 = r7945499 * r7945511;
        double r7945513 = r7945510 / r7945512;
        double r7945514 = 1.8091015183831773e+43;
        bool r7945515 = r7945494 <= r7945514;
        double r7945516 = 1.0;
        double r7945517 = r7945509 + r7945494;
        double r7945518 = r7945516 / r7945517;
        double r7945519 = 0.5;
        double r7945520 = sqrt(r7945519);
        double r7945521 = -0.25;
        double r7945522 = r7945497 / r7945521;
        double r7945523 = r7945522 * r7945520;
        double r7945524 = r7945520 * r7945523;
        double r7945525 = r7945518 * r7945524;
        double r7945526 = -r7945498;
        double r7945527 = r7945515 ? r7945525 : r7945526;
        double r7945528 = r7945503 ? r7945513 : r7945527;
        double r7945529 = r7945496 ? r7945501 : r7945528;
        return r7945529;
}

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

Original33.9
Target20.9
Herbie7.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.157094219357017e+135

    1. Initial program 54.4

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

    2. Simplified54.4

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

    3. Taylor expanded around -inf 2.8

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

    if -3.157094219357017e+135 < b < 9.088113400659685e-185

    1. Initial program 10.8

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

    2. Simplified10.8

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

    if 9.088113400659685e-185 < b < 1.8091015183831773e+43

    1. Initial program 34.3

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

    2. Simplified34.4

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

    4. Applied clear-num34.4

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

    6. Applied flip--34.5

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

    7. Applied associate-/r/34.6

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

    8. Applied *-un-lft-identity34.6

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

    9. Applied times-frac34.6

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

    10. Simplified17.1

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

    12. Applied add-sqr-sqrt17.6

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

    13. Applied *-un-lft-identity17.6

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

    14. Applied times-frac17.5

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

    15. Applied *-un-lft-identity17.5

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

    16. Applied times-frac17.4

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

    17. Simplified17.4

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

    18. Simplified7.7

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

    if 1.8091015183831773e+43 < b

    1. Initial program 56.4

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

    2. Simplified56.4

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

    3. Taylor expanded around inf 4.2

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

    4. Simplified4.2

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

  4. Final simplification7.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.157094219357017 \cdot 10^{+135}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 9.088113400659685 \cdot 10^{-185}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 1.8091015183831773 \cdot 10^{+43}:\\ \;\;\;\;\frac{1}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + b} \cdot \left(\sqrt{\frac{1}{2}} \cdot \left(\frac{c}{\frac{-1}{4}} \cdot \sqrt{\frac{1}{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))