Average Error: 34.5 → 10.1
Time: 7.1s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.6260438117910197 \cdot 10^{21}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.16764411094466422 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right) + b \cdot \left(b - b\right)}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le -5.52775192595066085 \cdot 10^{-141}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.3356876929369832 \cdot 10^{53}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]
\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 -1.6260438117910197 \cdot 10^{21}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -1.16764411094466422 \cdot 10^{-83}:\\
\;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right) + b \cdot \left(b - b\right)}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}}{2 \cdot a}\\

\mathbf{elif}\;b \le -5.52775192595066085 \cdot 10^{-141}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 3.3356876929369832 \cdot 10^{53}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r77738 = b;
        double r77739 = -r77738;
        double r77740 = r77738 * r77738;
        double r77741 = 4.0;
        double r77742 = a;
        double r77743 = c;
        double r77744 = r77742 * r77743;
        double r77745 = r77741 * r77744;
        double r77746 = r77740 - r77745;
        double r77747 = sqrt(r77746);
        double r77748 = r77739 - r77747;
        double r77749 = 2.0;
        double r77750 = r77749 * r77742;
        double r77751 = r77748 / r77750;
        return r77751;
}
↓
double f(double a, double b, double c) {
        double r77752 = b;
        double r77753 = -1.6260438117910197e+21;
        bool r77754 = r77752 <= r77753;
        double r77755 = -1.0;
        double r77756 = c;
        double r77757 = r77756 / r77752;
        double r77758 = r77755 * r77757;
        double r77759 = -1.1676441109446642e-83;
        bool r77760 = r77752 <= r77759;
        double r77761 = 4.0;
        double r77762 = a;
        double r77763 = r77762 * r77756;
        double r77764 = r77761 * r77763;
        double r77765 = r77752 - r77752;
        double r77766 = r77752 * r77765;
        double r77767 = r77764 + r77766;
        double r77768 = -r77764;
        double r77769 = fma(r77752, r77752, r77768);
        double r77770 = sqrt(r77769);
        double r77771 = r77770 - r77752;
        double r77772 = r77767 / r77771;
        double r77773 = 2.0;
        double r77774 = r77773 * r77762;
        double r77775 = r77772 / r77774;
        double r77776 = -5.527751925950661e-141;
        bool r77777 = r77752 <= r77776;
        double r77778 = 3.3356876929369832e+53;
        bool r77779 = r77752 <= r77778;
        double r77780 = -r77752;
        double r77781 = r77780 - r77770;
        double r77782 = r77781 / r77774;
        double r77783 = r77752 / r77762;
        double r77784 = r77755 * r77783;
        double r77785 = r77779 ? r77782 : r77784;
        double r77786 = r77777 ? r77758 : r77785;
        double r77787 = r77760 ? r77775 : r77786;
        double r77788 = r77754 ? r77758 : r77787;
        return r77788;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.5
Target20.8
Herbie10.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.6260438117910197e+21 or -1.1676441109446642e-83 < b < -5.527751925950661e-141

    1. Initial program 52.7

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

    2. Taylor expanded around -inf 9.4

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

    if -1.6260438117910197e+21 < b < -1.1676441109446642e-83

    1. Initial program 39.4

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

    3. Applied clear-num39.4

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

    5. Applied *-un-lft-identity39.4

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

    6. Applied add-cube-cbrt39.4

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot \frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

    7. Applied times-frac39.4

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}\]

    8. Simplified39.4

      \[\leadsto \color{blue}{1} \cdot \frac{\sqrt[3]{1}}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

    9. Simplified39.4

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

    11. Applied flip--39.4

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

    12. Simplified16.7

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

    13. Simplified16.7

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

    if -5.527751925950661e-141 < b < 3.3356876929369832e+53

    1. Initial program 12.0

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

    3. Applied clear-num12.1

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

    5. Applied *-un-lft-identity12.1

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

    6. Applied add-cube-cbrt12.1

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot \frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

    7. Applied times-frac12.1

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}\]

    8. Simplified12.1

      \[\leadsto \color{blue}{1} \cdot \frac{\sqrt[3]{1}}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

    9. Simplified12.0

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

    if 3.3356876929369832e+53 < b

    1. Initial program 38.5

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

    3. Applied clear-num38.6

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

    4. Taylor expanded around 0 5.3

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.6260438117910197 \cdot 10^{21}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.16764411094466422 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right) + b \cdot \left(b - b\right)}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le -5.52775192595066085 \cdot 10^{-141}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.3356876929369832 \cdot 10^{53}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))