Average Error: 20.0 → 7.6
Time: 37.1s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.024722505568394940568276880731248223192 \cdot 10^{88}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(\frac{\left(2 \cdot c\right) \cdot a}{b} - b\right)}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \le 1.35963024204659672399001542173939626506 \cdot 10^{143}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{\sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c}}} \cdot \sqrt{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}} + \left(-b\right)}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(\frac{a}{b} \cdot c, 2, -2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}{a \cdot 2}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le -5.024722505568394940568276880731248223192 \cdot 10^{88}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b\right) + \left(\frac{\left(2 \cdot c\right) \cdot a}{b} - b\right)}{a \cdot 2}\\

\end{array}\\

\mathbf{elif}\;b \le 1.35963024204659672399001542173939626506 \cdot 10^{143}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{\sqrt{\sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c}}} \cdot \sqrt{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}} + \left(-b\right)}{a \cdot 2}\\

\end{array}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(\frac{a}{b} \cdot c, 2, -2 \cdot b\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}{a \cdot 2}\\

\end{array}
double f(double a, double b, double c) {
        double r1132661 = b;
        double r1132662 = 0.0;
        bool r1132663 = r1132661 >= r1132662;
        double r1132664 = 2.0;
        double r1132665 = c;
        double r1132666 = r1132664 * r1132665;
        double r1132667 = -r1132661;
        double r1132668 = r1132661 * r1132661;
        double r1132669 = 4.0;
        double r1132670 = a;
        double r1132671 = r1132669 * r1132670;
        double r1132672 = r1132671 * r1132665;
        double r1132673 = r1132668 - r1132672;
        double r1132674 = sqrt(r1132673);
        double r1132675 = r1132667 - r1132674;
        double r1132676 = r1132666 / r1132675;
        double r1132677 = r1132667 + r1132674;
        double r1132678 = r1132664 * r1132670;
        double r1132679 = r1132677 / r1132678;
        double r1132680 = r1132663 ? r1132676 : r1132679;
        return r1132680;
}
↓
double f(double a, double b, double c) {
        double r1132681 = b;
        double r1132682 = -5.024722505568395e+88;
        bool r1132683 = r1132681 <= r1132682;
        double r1132684 = 0.0;
        bool r1132685 = r1132681 >= r1132684;
        double r1132686 = 2.0;
        double r1132687 = c;
        double r1132688 = r1132686 * r1132687;
        double r1132689 = -r1132681;
        double r1132690 = r1132681 * r1132681;
        double r1132691 = a;
        double r1132692 = 4.0;
        double r1132693 = r1132691 * r1132692;
        double r1132694 = r1132693 * r1132687;
        double r1132695 = r1132690 - r1132694;
        double r1132696 = sqrt(r1132695);
        double r1132697 = r1132689 - r1132696;
        double r1132698 = r1132688 / r1132697;
        double r1132699 = r1132688 * r1132691;
        double r1132700 = r1132699 / r1132681;
        double r1132701 = r1132700 - r1132681;
        double r1132702 = r1132689 + r1132701;
        double r1132703 = r1132691 * r1132686;
        double r1132704 = r1132702 / r1132703;
        double r1132705 = r1132685 ? r1132698 : r1132704;
        double r1132706 = 1.3596302420465967e+143;
        bool r1132707 = r1132681 <= r1132706;
        double r1132708 = cbrt(r1132695);
        double r1132709 = sqrt(r1132708);
        double r1132710 = r1132708 * r1132708;
        double r1132711 = sqrt(r1132710);
        double r1132712 = r1132709 * r1132711;
        double r1132713 = sqrt(r1132712);
        double r1132714 = sqrt(r1132696);
        double r1132715 = r1132713 * r1132714;
        double r1132716 = r1132715 + r1132689;
        double r1132717 = r1132716 / r1132703;
        double r1132718 = r1132685 ? r1132698 : r1132717;
        double r1132719 = r1132691 / r1132681;
        double r1132720 = r1132719 * r1132687;
        double r1132721 = -2.0;
        double r1132722 = r1132721 * r1132681;
        double r1132723 = fma(r1132720, r1132686, r1132722);
        double r1132724 = r1132688 / r1132723;
        double r1132725 = r1132696 + r1132689;
        double r1132726 = r1132725 / r1132703;
        double r1132727 = r1132685 ? r1132724 : r1132726;
        double r1132728 = r1132707 ? r1132718 : r1132727;
        double r1132729 = r1132683 ? r1132705 : r1132728;
        return r1132729;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -5.024722505568395e+88

    1. Initial program 44.7

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

    2. Taylor expanded around -inf 10.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(2 \cdot \frac{a \cdot c}{b} - b\right)}{2 \cdot a}\\ \end{array}\]

    3. Simplified10.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(\frac{a \cdot \left(2 \cdot c\right)}{b} - b\right)}{2 \cdot a}\\ \end{array}\]

    if -5.024722505568395e+88 < b < 1.3596302420465967e+143

    1. Initial program 8.6

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt8.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]

    4. Applied sqrt-prod8.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt8.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{\left(\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\\ \end{array}\]

    7. Applied sqrt-prod8.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\\ \end{array}\]

    if 1.3596302420465967e+143 < b

    1. Initial program 36.6

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

    2. Taylor expanded around inf 6.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    3. Simplified1.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\mathsf{fma}\left(\frac{a}{b} \cdot c, 2, -2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification7.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.024722505568394940568276880731248223192 \cdot 10^{88}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(\frac{\left(2 \cdot c\right) \cdot a}{b} - b\right)}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \le 1.35963024204659672399001542173939626506 \cdot 10^{143}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{\sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c} \cdot \sqrt[3]{b \cdot b - \left(a \cdot 4\right) \cdot c}}} \cdot \sqrt{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}} + \left(-b\right)}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(\frac{a}{b} \cdot c, 2, -2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}{a \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 2"
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))