Average Error: 33.5 → 10.0
Time: 7.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.0323767944871679 \cdot 10^{127}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.17528679488360856 \cdot 10^{-69}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -4.0323767944871679 \cdot 10^{127}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.17528679488360856 \cdot 10^{-69}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r181453 = b;
        double r181454 = -r181453;
        double r181455 = r181453 * r181453;
        double r181456 = 4.0;
        double r181457 = a;
        double r181458 = r181456 * r181457;
        double r181459 = c;
        double r181460 = r181458 * r181459;
        double r181461 = r181455 - r181460;
        double r181462 = sqrt(r181461);
        double r181463 = r181454 + r181462;
        double r181464 = 2.0;
        double r181465 = r181464 * r181457;
        double r181466 = r181463 / r181465;
        return r181466;
}
↓
double f(double a, double b, double c) {
        double r181467 = b;
        double r181468 = -4.032376794487168e+127;
        bool r181469 = r181467 <= r181468;
        double r181470 = 1.0;
        double r181471 = c;
        double r181472 = r181471 / r181467;
        double r181473 = a;
        double r181474 = r181467 / r181473;
        double r181475 = r181472 - r181474;
        double r181476 = r181470 * r181475;
        double r181477 = 1.1752867948836086e-69;
        bool r181478 = r181467 <= r181477;
        double r181479 = -r181467;
        double r181480 = r181467 * r181467;
        double r181481 = 4.0;
        double r181482 = r181481 * r181473;
        double r181483 = r181482 * r181471;
        double r181484 = r181480 - r181483;
        double r181485 = sqrt(r181484);
        double r181486 = r181479 + r181485;
        double r181487 = 2.0;
        double r181488 = r181487 * r181473;
        double r181489 = r181486 / r181488;
        double r181490 = -1.0;
        double r181491 = r181490 * r181472;
        double r181492 = r181478 ? r181489 : r181491;
        double r181493 = r181469 ? r181476 : r181492;
        return r181493;
}

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

Original33.5
Target20.6
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.032376794487168e+127

    1. Initial program 53.1

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

    2. Taylor expanded around -inf 3.1

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

    3. Simplified3.1

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

    if -4.032376794487168e+127 < b < 1.1752867948836086e-69

    1. Initial program 12.7

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

    if 1.1752867948836086e-69 < b

    1. Initial program 53.9

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

    2. Taylor expanded around inf 8.8

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.0323767944871679 \cdot 10^{127}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.17528679488360856 \cdot 10^{-69}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(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)))