Average Error: 34.2 → 10.4
Time: 12.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.12310353364421125 \cdot 10^{95}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 3.446447862996811 \cdot 10^{-75}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -4.12310353364421125 \cdot 10^{95}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 3.446447862996811 \cdot 10^{-75}:\\
\;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b\right) \cdot \frac{\frac{1}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r102700 = b;
        double r102701 = -r102700;
        double r102702 = r102700 * r102700;
        double r102703 = 4.0;
        double r102704 = a;
        double r102705 = c;
        double r102706 = r102704 * r102705;
        double r102707 = r102703 * r102706;
        double r102708 = r102702 - r102707;
        double r102709 = sqrt(r102708);
        double r102710 = r102701 + r102709;
        double r102711 = 2.0;
        double r102712 = r102711 * r102704;
        double r102713 = r102710 / r102712;
        return r102713;
}
↓
double f(double a, double b, double c) {
        double r102714 = b;
        double r102715 = -4.123103533644211e+95;
        bool r102716 = r102714 <= r102715;
        double r102717 = 1.0;
        double r102718 = c;
        double r102719 = r102718 / r102714;
        double r102720 = a;
        double r102721 = r102714 / r102720;
        double r102722 = r102719 - r102721;
        double r102723 = r102717 * r102722;
        double r102724 = 3.446447862996811e-75;
        bool r102725 = r102714 <= r102724;
        double r102726 = 4.0;
        double r102727 = r102720 * r102718;
        double r102728 = r102726 * r102727;
        double r102729 = -r102728;
        double r102730 = fma(r102714, r102714, r102729);
        double r102731 = sqrt(r102730);
        double r102732 = r102731 - r102714;
        double r102733 = 1.0;
        double r102734 = 2.0;
        double r102735 = r102733 / r102734;
        double r102736 = r102735 / r102720;
        double r102737 = r102732 * r102736;
        double r102738 = -1.0;
        double r102739 = r102738 * r102719;
        double r102740 = r102725 ? r102737 : r102739;
        double r102741 = r102716 ? r102723 : r102740;
        return r102741;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.2
Target21.1
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.123103533644211e+95

    1. Initial program 47.3

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

    2. Simplified47.3

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

    3. Taylor expanded around -inf 3.8

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

    4. Simplified3.8

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

    if -4.123103533644211e+95 < b < 3.446447862996811e-75

    1. Initial program 13.3

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

    2. Simplified13.3

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

    4. Applied *-un-lft-identity13.3

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

    5. Applied div-inv13.3

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

    6. Applied times-frac13.4

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

    7. Simplified13.4

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

    if 3.446447862996811e-75 < b

    1. Initial program 52.5

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

    2. Simplified52.5

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

    3. Taylor expanded around inf 9.7

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.12310353364421125 \cdot 10^{95}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 3.446447862996811 \cdot 10^{-75}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +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)))