Average Error: 34.8 → 10.7
Time: 4.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.60514141786167054 \cdot 10^{33}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.92049775718538 \cdot 10^{-66}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -4.60514141786167054 \cdot 10^{33}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.92049775718538 \cdot 10^{-66}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r91604 = b;
        double r91605 = -r91604;
        double r91606 = r91604 * r91604;
        double r91607 = 4.0;
        double r91608 = a;
        double r91609 = r91607 * r91608;
        double r91610 = c;
        double r91611 = r91609 * r91610;
        double r91612 = r91606 - r91611;
        double r91613 = sqrt(r91612);
        double r91614 = r91605 + r91613;
        double r91615 = 2.0;
        double r91616 = r91615 * r91608;
        double r91617 = r91614 / r91616;
        return r91617;
}
↓
double f(double a, double b, double c) {
        double r91618 = b;
        double r91619 = -4.6051414178616705e+33;
        bool r91620 = r91618 <= r91619;
        double r91621 = 1.0;
        double r91622 = c;
        double r91623 = r91622 / r91618;
        double r91624 = a;
        double r91625 = r91618 / r91624;
        double r91626 = r91623 - r91625;
        double r91627 = r91621 * r91626;
        double r91628 = 1.92049775718538e-66;
        bool r91629 = r91618 <= r91628;
        double r91630 = -r91618;
        double r91631 = r91618 * r91618;
        double r91632 = 4.0;
        double r91633 = r91632 * r91624;
        double r91634 = r91633 * r91622;
        double r91635 = r91631 - r91634;
        double r91636 = sqrt(r91635);
        double r91637 = r91630 + r91636;
        double r91638 = 1.0;
        double r91639 = 2.0;
        double r91640 = r91639 * r91624;
        double r91641 = r91638 / r91640;
        double r91642 = r91637 * r91641;
        double r91643 = -1.0;
        double r91644 = r91643 * r91623;
        double r91645 = r91629 ? r91642 : r91644;
        double r91646 = r91620 ? r91627 : r91645;
        return r91646;
}

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

Original34.8
Target21.2
Herbie10.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.6051414178616705e+33

    1. Initial program 36.3

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

    2. Taylor expanded around -inf 6.9

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

    3. Simplified6.9

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

    if -4.6051414178616705e+33 < b < 1.92049775718538e-66

    1. Initial program 15.2

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

    3. Applied div-inv15.3

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

    if 1.92049775718538e-66 < b

    1. Initial program 54.0

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

    2. Taylor expanded around inf 8.1

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

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.60514141786167054 \cdot 10^{33}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.92049775718538 \cdot 10^{-66}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(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)))