Average Error: 34.1 → 9.2
Time: 5.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.3044033969831823 \cdot 10^{153}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.9238883452280037 \cdot 10^{-130}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \left(-b\right)}{2 \cdot a}\\ \mathbf{elif}\;b \le 4.01993084419163312 \cdot 10^{109}:\\ \;\;\;\;\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.3044033969831823 \cdot 10^{153}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.9238883452280037 \cdot 10^{-130}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \left(-b\right)}{2 \cdot a}\\

\mathbf{elif}\;b \le 4.01993084419163312 \cdot 10^{109}:\\
\;\;\;\;\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - 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 r74572 = b;
        double r74573 = -r74572;
        double r74574 = r74572 * r74572;
        double r74575 = 4.0;
        double r74576 = a;
        double r74577 = c;
        double r74578 = r74576 * r74577;
        double r74579 = r74575 * r74578;
        double r74580 = r74574 - r74579;
        double r74581 = sqrt(r74580);
        double r74582 = r74573 + r74581;
        double r74583 = 2.0;
        double r74584 = r74583 * r74576;
        double r74585 = r74582 / r74584;
        return r74585;
}
↓
double f(double a, double b, double c) {
        double r74586 = b;
        double r74587 = -2.3044033969831823e+153;
        bool r74588 = r74586 <= r74587;
        double r74589 = 1.0;
        double r74590 = c;
        double r74591 = r74590 / r74586;
        double r74592 = a;
        double r74593 = r74586 / r74592;
        double r74594 = r74591 - r74593;
        double r74595 = r74589 * r74594;
        double r74596 = 1.9238883452280037e-130;
        bool r74597 = r74586 <= r74596;
        double r74598 = r74586 * r74586;
        double r74599 = 4.0;
        double r74600 = r74592 * r74590;
        double r74601 = r74599 * r74600;
        double r74602 = r74598 - r74601;
        double r74603 = sqrt(r74602);
        double r74604 = -r74586;
        double r74605 = r74603 + r74604;
        double r74606 = 2.0;
        double r74607 = r74606 * r74592;
        double r74608 = r74605 / r74607;
        double r74609 = 4.019930844191633e+109;
        bool r74610 = r74586 <= r74609;
        double r74611 = 0.0;
        double r74612 = r74611 + r74601;
        double r74613 = r74604 - r74603;
        double r74614 = r74612 / r74613;
        double r74615 = r74614 / r74607;
        double r74616 = -1.0;
        double r74617 = r74616 * r74591;
        double r74618 = r74610 ? r74615 : r74617;
        double r74619 = r74597 ? r74608 : r74618;
        double r74620 = r74588 ? r74595 : r74619;
        return r74620;
}

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

Original34.1
Target20.6
Herbie9.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.3044033969831823e+153

    1. Initial program 63.5

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

    2. Taylor expanded around -inf 2.0

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

    3. Simplified2.0

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

    if -2.3044033969831823e+153 < b < 1.9238883452280037e-130

    1. Initial program 11.3

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

    3. Applied +-commutative11.3

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

    if 1.9238883452280037e-130 < b < 4.019930844191633e+109

    1. Initial program 40.3

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

    3. Applied flip-+40.3

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

    4. Simplified15.5

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

    if 4.019930844191633e+109 < b

    1. Initial program 59.9

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

    2. Taylor expanded around inf 2.4

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

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.3044033969831823 \cdot 10^{153}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.9238883452280037 \cdot 10^{-130}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \left(-b\right)}{2 \cdot a}\\ \mathbf{elif}\;b \le 4.01993084419163312 \cdot 10^{109}:\\ \;\;\;\;\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :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)))