Average Error: 34.4 → 9.1
Time: 5.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.0461303908572575 \cdot 10^{65}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.98127510036099003 \cdot 10^{-264}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{elif}\;b \le 2114787851.2472425:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{2 \cdot a}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.0461303908572575 \cdot 10^{65}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 2.98127510036099003 \cdot 10^{-264}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\

\mathbf{elif}\;b \le 2114787851.2472425:\\
\;\;\;\;\frac{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{2 \cdot a}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r158473 = b;
        double r158474 = -r158473;
        double r158475 = r158473 * r158473;
        double r158476 = 4.0;
        double r158477 = a;
        double r158478 = r158476 * r158477;
        double r158479 = c;
        double r158480 = r158478 * r158479;
        double r158481 = r158475 - r158480;
        double r158482 = sqrt(r158481);
        double r158483 = r158474 + r158482;
        double r158484 = 2.0;
        double r158485 = r158484 * r158477;
        double r158486 = r158483 / r158485;
        return r158486;
}
↓
double f(double a, double b, double c) {
        double r158487 = b;
        double r158488 = -1.0461303908572575e+65;
        bool r158489 = r158487 <= r158488;
        double r158490 = 1.0;
        double r158491 = c;
        double r158492 = r158491 / r158487;
        double r158493 = a;
        double r158494 = r158487 / r158493;
        double r158495 = r158492 - r158494;
        double r158496 = r158490 * r158495;
        double r158497 = 2.98127510036099e-264;
        bool r158498 = r158487 <= r158497;
        double r158499 = 1.0;
        double r158500 = 2.0;
        double r158501 = r158500 * r158493;
        double r158502 = -r158487;
        double r158503 = r158487 * r158487;
        double r158504 = 4.0;
        double r158505 = r158504 * r158493;
        double r158506 = r158505 * r158491;
        double r158507 = r158503 - r158506;
        double r158508 = sqrt(r158507);
        double r158509 = r158502 + r158508;
        double r158510 = r158501 / r158509;
        double r158511 = r158499 / r158510;
        double r158512 = 2114787851.2472425;
        bool r158513 = r158487 <= r158512;
        double r158514 = 2.0;
        double r158515 = pow(r158487, r158514);
        double r158516 = r158515 - r158515;
        double r158517 = r158493 * r158491;
        double r158518 = r158504 * r158517;
        double r158519 = r158516 + r158518;
        double r158520 = r158519 / r158501;
        double r158521 = r158502 - r158508;
        double r158522 = r158520 / r158521;
        double r158523 = -1.0;
        double r158524 = r158523 * r158492;
        double r158525 = r158513 ? r158522 : r158524;
        double r158526 = r158498 ? r158511 : r158525;
        double r158527 = r158489 ? r158496 : r158526;
        return r158527;
}

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

Original34.4
Target21.0
Herbie9.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.0461303908572575e+65

    1. Initial program 41.3

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

    2. Taylor expanded around -inf 4.6

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

    3. Simplified4.6

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

    if -1.0461303908572575e+65 < b < 2.98127510036099e-264

    1. Initial program 10.7

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

    3. Applied clear-num10.8

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

    if 2.98127510036099e-264 < b < 2114787851.2472425

    1. Initial program 27.8

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

    3. Applied clear-num27.8

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

    5. Applied flip-+27.8

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

    6. Applied associate-/r/27.9

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

    7. Applied associate-/r*27.9

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

    8. Simplified17.7

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

    if 2114787851.2472425 < b

    1. Initial program 56.3

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

    2. Taylor expanded around inf 4.8

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.0461303908572575 \cdot 10^{65}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.98127510036099003 \cdot 10^{-264}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{elif}\;b \le 2114787851.2472425:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{2 \cdot a}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- 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)))