Average Error: 34.2 → 11.6
Time: 19.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.555632367828988861043913196266489993904 \cdot 10^{101}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.555632367828988861043913196266489993904 \cdot 10^{101}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r123448 = b;
        double r123449 = -r123448;
        double r123450 = r123448 * r123448;
        double r123451 = 4.0;
        double r123452 = a;
        double r123453 = r123451 * r123452;
        double r123454 = c;
        double r123455 = r123453 * r123454;
        double r123456 = r123450 - r123455;
        double r123457 = sqrt(r123456);
        double r123458 = r123449 + r123457;
        double r123459 = 2.0;
        double r123460 = r123459 * r123452;
        double r123461 = r123458 / r123460;
        return r123461;
}
↓
double f(double a, double b, double c) {
        double r123462 = b;
        double r123463 = -1.555632367828989e+101;
        bool r123464 = r123462 <= r123463;
        double r123465 = 1.0;
        double r123466 = c;
        double r123467 = r123466 / r123462;
        double r123468 = a;
        double r123469 = r123462 / r123468;
        double r123470 = r123467 - r123469;
        double r123471 = r123465 * r123470;
        double r123472 = 7.455592343308264e-170;
        bool r123473 = r123462 <= r123472;
        double r123474 = -r123462;
        double r123475 = 2.0;
        double r123476 = pow(r123462, r123475);
        double r123477 = 4.0;
        double r123478 = r123468 * r123466;
        double r123479 = r123477 * r123478;
        double r123480 = r123476 - r123479;
        double r123481 = sqrt(r123480);
        double r123482 = r123474 + r123481;
        double r123483 = 2.0;
        double r123484 = r123483 * r123468;
        double r123485 = r123482 / r123484;
        double r123486 = -1.0;
        double r123487 = r123486 * r123467;
        double r123488 = r123473 ? r123485 : r123487;
        double r123489 = r123464 ? r123471 : r123488;
        return r123489;
}

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

Original34.2
Target20.8
Herbie11.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.555632367828989e+101

    1. Initial program 47.4

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

    2. Taylor expanded around -inf 3.6

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

    3. Simplified3.6

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

    if -1.555632367828989e+101 < b < 7.455592343308264e-170

    1. Initial program 11.7

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

    2. Taylor expanded around 0 11.7

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

    if 7.455592343308264e-170 < b

    1. Initial program 48.9

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

    2. Taylor expanded around inf 14.1

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

  4. Final simplification11.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.555632367828988861043913196266489993904 \cdot 10^{101}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(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)))