Average Error: 34.3 → 8.6
Time: 30.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9661478263987.111328125:\\ \;\;\;\;\frac{c \cdot \sqrt[3]{-1}}{b} \cdot \left(\sqrt[3]{-1} \cdot \sqrt[3]{-1}\right)\\ \mathbf{elif}\;b \le -1.244932636718084290671504385697528170065 \cdot 10^{-183}:\\ \;\;\;\;\frac{\frac{a \cdot \left(c \cdot 4\right) + \left({b}^{2} - {b}^{2}\right)}{\sqrt{\mathsf{fma}\left(a, 4 \cdot \left(-c\right), {b}^{2}\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le 2.280923374767716130571300401308257426651 \cdot 10^{83}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{\mathsf{fma}\left(a, 4 \cdot \left(-c\right), {b}^{2}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -9661478263987.111328125:\\
\;\;\;\;\frac{c \cdot \sqrt[3]{-1}}{b} \cdot \left(\sqrt[3]{-1} \cdot \sqrt[3]{-1}\right)\\

\mathbf{elif}\;b \le -1.244932636718084290671504385697528170065 \cdot 10^{-183}:\\
\;\;\;\;\frac{\frac{a \cdot \left(c \cdot 4\right) + \left({b}^{2} - {b}^{2}\right)}{\sqrt{\mathsf{fma}\left(a, 4 \cdot \left(-c\right), {b}^{2}\right)} - b}}{2 \cdot a}\\

\mathbf{elif}\;b \le 2.280923374767716130571300401308257426651 \cdot 10^{83}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{\mathsf{fma}\left(a, 4 \cdot \left(-c\right), {b}^{2}\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r62835 = b;
        double r62836 = -r62835;
        double r62837 = r62835 * r62835;
        double r62838 = 4.0;
        double r62839 = a;
        double r62840 = c;
        double r62841 = r62839 * r62840;
        double r62842 = r62838 * r62841;
        double r62843 = r62837 - r62842;
        double r62844 = sqrt(r62843);
        double r62845 = r62836 - r62844;
        double r62846 = 2.0;
        double r62847 = r62846 * r62839;
        double r62848 = r62845 / r62847;
        return r62848;
}
↓
double f(double a, double b, double c) {
        double r62849 = b;
        double r62850 = -9661478263987.111;
        bool r62851 = r62849 <= r62850;
        double r62852 = c;
        double r62853 = -1.0;
        double r62854 = cbrt(r62853);
        double r62855 = r62852 * r62854;
        double r62856 = r62855 / r62849;
        double r62857 = r62854 * r62854;
        double r62858 = r62856 * r62857;
        double r62859 = -1.2449326367180843e-183;
        bool r62860 = r62849 <= r62859;
        double r62861 = a;
        double r62862 = 4.0;
        double r62863 = r62852 * r62862;
        double r62864 = r62861 * r62863;
        double r62865 = 2.0;
        double r62866 = pow(r62849, r62865);
        double r62867 = r62866 - r62866;
        double r62868 = r62864 + r62867;
        double r62869 = -r62852;
        double r62870 = r62862 * r62869;
        double r62871 = fma(r62861, r62870, r62866);
        double r62872 = sqrt(r62871);
        double r62873 = r62872 - r62849;
        double r62874 = r62868 / r62873;
        double r62875 = 2.0;
        double r62876 = r62875 * r62861;
        double r62877 = r62874 / r62876;
        double r62878 = 2.280923374767716e+83;
        bool r62879 = r62849 <= r62878;
        double r62880 = -r62849;
        double r62881 = r62880 / r62876;
        double r62882 = r62872 / r62876;
        double r62883 = r62881 - r62882;
        double r62884 = 1.0;
        double r62885 = r62852 / r62849;
        double r62886 = r62849 / r62861;
        double r62887 = r62885 - r62886;
        double r62888 = r62884 * r62887;
        double r62889 = r62879 ? r62883 : r62888;
        double r62890 = r62860 ? r62877 : r62889;
        double r62891 = r62851 ? r62858 : r62890;
        return r62891;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.3
Target21.1
Herbie8.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -9661478263987.111

    1. Initial program 56.4

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

    2. Taylor expanded around -inf 5.0

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

    3. Simplified5.9

      \[\leadsto \color{blue}{\frac{-1}{\frac{b}{c}}}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity5.9

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

    6. Applied *-un-lft-identity5.9

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

    7. Applied times-frac5.9

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

    8. Applied add-cube-cbrt5.9

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

    9. Applied times-frac5.9

      \[\leadsto \color{blue}{\frac{\sqrt[3]{-1} \cdot \sqrt[3]{-1}}{\frac{1}{1}} \cdot \frac{\sqrt[3]{-1}}{\frac{b}{c}}}\]

    10. Simplified5.9

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

    11. Simplified5.0

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

    if -9661478263987.111 < b < -1.2449326367180843e-183

    1. Initial program 34.1

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

    3. Applied flip--34.1

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

    4. Simplified18.3

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

    5. Simplified18.3

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

    if -1.2449326367180843e-183 < b < 2.280923374767716e+83

    1. Initial program 10.5

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

    3. Applied div-sub10.5

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

    4. Simplified10.5

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

    5. Simplified10.5

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

    if 2.280923374767716e+83 < b

    1. Initial program 43.8

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

    2. Taylor expanded around inf 3.5

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

    3. Simplified3.5

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9661478263987.111328125:\\ \;\;\;\;\frac{c \cdot \sqrt[3]{-1}}{b} \cdot \left(\sqrt[3]{-1} \cdot \sqrt[3]{-1}\right)\\ \mathbf{elif}\;b \le -1.244932636718084290671504385697528170065 \cdot 10^{-183}:\\ \;\;\;\;\frac{\frac{a \cdot \left(c \cdot 4\right) + \left({b}^{2} - {b}^{2}\right)}{\sqrt{\mathsf{fma}\left(a, 4 \cdot \left(-c\right), {b}^{2}\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le 2.280923374767716130571300401308257426651 \cdot 10^{83}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{\mathsf{fma}\left(a, 4 \cdot \left(-c\right), {b}^{2}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))