Average Error: 34.5 → 6.4
Time: 7.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.52306947897632778228201833866057110671 \cdot 10^{123}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -1.310125818367260256329730829931519203575 \cdot 10^{-294}:\\ \;\;\;\;\frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{\frac{2 \cdot a}{\sqrt{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}\\ \mathbf{elif}\;b \le 1.076384822638075226373023800566186486206 \cdot 10^{126}:\\ \;\;\;\;\frac{1}{\frac{0.5}{c} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -5.52306947897632778228201833866057110671 \cdot 10^{123}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -1.310125818367260256329730829931519203575 \cdot 10^{-294}:\\
\;\;\;\;\frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{\frac{2 \cdot a}{\sqrt{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}\\

\mathbf{elif}\;b \le 1.076384822638075226373023800566186486206 \cdot 10^{126}:\\
\;\;\;\;\frac{1}{\frac{0.5}{c} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\\

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

\end{array}
double f(double a, double b, double c) {
        double r137579 = b;
        double r137580 = -r137579;
        double r137581 = r137579 * r137579;
        double r137582 = 4.0;
        double r137583 = a;
        double r137584 = c;
        double r137585 = r137583 * r137584;
        double r137586 = r137582 * r137585;
        double r137587 = r137581 - r137586;
        double r137588 = sqrt(r137587);
        double r137589 = r137580 + r137588;
        double r137590 = 2.0;
        double r137591 = r137590 * r137583;
        double r137592 = r137589 / r137591;
        return r137592;
}
↓
double f(double a, double b, double c) {
        double r137593 = b;
        double r137594 = -5.523069478976328e+123;
        bool r137595 = r137593 <= r137594;
        double r137596 = 1.0;
        double r137597 = c;
        double r137598 = r137597 / r137593;
        double r137599 = a;
        double r137600 = r137593 / r137599;
        double r137601 = r137598 - r137600;
        double r137602 = r137596 * r137601;
        double r137603 = -1.3101258183672603e-294;
        bool r137604 = r137593 <= r137603;
        double r137605 = -r137593;
        double r137606 = r137593 * r137593;
        double r137607 = 4.0;
        double r137608 = r137599 * r137597;
        double r137609 = r137607 * r137608;
        double r137610 = r137606 - r137609;
        double r137611 = sqrt(r137610);
        double r137612 = r137605 + r137611;
        double r137613 = sqrt(r137612);
        double r137614 = 2.0;
        double r137615 = r137614 * r137599;
        double r137616 = r137615 / r137613;
        double r137617 = r137613 / r137616;
        double r137618 = 1.0763848226380752e+126;
        bool r137619 = r137593 <= r137618;
        double r137620 = 1.0;
        double r137621 = 0.5;
        double r137622 = r137621 / r137597;
        double r137623 = r137605 - r137611;
        double r137624 = r137622 * r137623;
        double r137625 = r137620 / r137624;
        double r137626 = -1.0;
        double r137627 = r137626 * r137598;
        double r137628 = r137619 ? r137625 : r137627;
        double r137629 = r137604 ? r137617 : r137628;
        double r137630 = r137595 ? r137602 : r137629;
        return r137630;
}

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

Original34.5
Target21.5
Herbie6.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -5.523069478976328e+123

    1. Initial program 54.0

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

    2. Taylor expanded around -inf 2.9

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

    3. Simplified2.9

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

    if -5.523069478976328e+123 < b < -1.3101258183672603e-294

    1. Initial program 8.1

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

    3. Applied add-sqr-sqrt8.5

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

    4. Applied associate-/l*8.5

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

    if -1.3101258183672603e-294 < b < 1.0763848226380752e+126

    1. Initial program 33.5

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

    3. Applied flip-+33.5

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

    4. Simplified16.4

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

    6. Applied clear-num16.6

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}}\]

    7. Simplified15.4

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

    8. Taylor expanded around 0 9.0

      \[\leadsto \frac{1}{\color{blue}{\frac{0.5}{c}} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\]

    if 1.0763848226380752e+126 < b

    1. Initial program 60.9

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

    2. Taylor expanded around inf 1.8

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.52306947897632778228201833866057110671 \cdot 10^{123}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -1.310125818367260256329730829931519203575 \cdot 10^{-294}:\\ \;\;\;\;\frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{\frac{2 \cdot a}{\sqrt{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}\\ \mathbf{elif}\;b \le 1.076384822638075226373023800566186486206 \cdot 10^{126}:\\ \;\;\;\;\frac{1}{\frac{0.5}{c} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019318 
(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)))