Average Error: 18.6 → 12.6
Time: 34.1s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 2.01827746078287 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;-\frac{\frac{\mathsf{fma}\left(b, b \cdot b, \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}\right)}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} - b, \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}, b \cdot b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{\sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)} - b}{2}}\\ \end{array}\\ \mathbf{elif}\;b \le 2.559678284282607 \cdot 10^{+69}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{2}}{c} \cdot \left(\sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)} - b\right)}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

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

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le 2.01827746078287 \cdot 10^{-310}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;-\frac{\frac{\mathsf{fma}\left(b, b \cdot b, \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}\right)}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} - b, \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}, b \cdot b\right)}}{a \cdot 2}\\

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

\end{array}\\

\mathbf{elif}\;b \le 2.559678284282607 \cdot 10^{+69}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{0}\\

\end{array}\\

\mathbf{elif}\;b \ge 0:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{\frac{1}{2}}{c} \cdot \left(\sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)} - b\right)}\\

\end{array}
double f(double a, double b, double c) {
        double r751926 = b;
        double r751927 = 0.0;
        bool r751928 = r751926 >= r751927;
        double r751929 = -r751926;
        double r751930 = r751926 * r751926;
        double r751931 = 4.0;
        double r751932 = a;
        double r751933 = r751931 * r751932;
        double r751934 = c;
        double r751935 = r751933 * r751934;
        double r751936 = r751930 - r751935;
        double r751937 = sqrt(r751936);
        double r751938 = r751929 - r751937;
        double r751939 = 2.0;
        double r751940 = r751939 * r751932;
        double r751941 = r751938 / r751940;
        double r751942 = r751939 * r751934;
        double r751943 = r751929 + r751937;
        double r751944 = r751942 / r751943;
        double r751945 = r751928 ? r751941 : r751944;
        return r751945;
}
↓
double f(double a, double b, double c) {
        double r751946 = b;
        double r751947 = 2.01827746078287e-310;
        bool r751948 = r751946 <= r751947;
        double r751949 = 0.0;
        bool r751950 = r751946 >= r751949;
        double r751951 = r751946 * r751946;
        double r751952 = a;
        double r751953 = c;
        double r751954 = r751952 * r751953;
        double r751955 = -4.0;
        double r751956 = fma(r751954, r751955, r751951);
        double r751957 = sqrt(r751956);
        double r751958 = r751956 * r751957;
        double r751959 = fma(r751946, r751951, r751958);
        double r751960 = r751957 - r751946;
        double r751961 = fma(r751960, r751957, r751951);
        double r751962 = r751959 / r751961;
        double r751963 = 2.0;
        double r751964 = r751952 * r751963;
        double r751965 = r751962 / r751964;
        double r751966 = -r751965;
        double r751967 = r751955 * r751952;
        double r751968 = fma(r751953, r751967, r751951);
        double r751969 = sqrt(r751968);
        double r751970 = r751969 - r751946;
        double r751971 = r751970 / r751963;
        double r751972 = r751953 / r751971;
        double r751973 = r751950 ? r751966 : r751972;
        double r751974 = 2.559678284282607e+69;
        bool r751975 = r751946 <= r751974;
        double r751976 = -r751946;
        double r751977 = r751976 - r751969;
        double r751978 = r751977 / r751964;
        double r751979 = r751953 / r751949;
        double r751980 = r751950 ? r751978 : r751979;
        double r751981 = r751953 / r751946;
        double r751982 = r751946 / r751952;
        double r751983 = r751981 - r751982;
        double r751984 = 1.0;
        double r751985 = 0.5;
        double r751986 = r751985 / r751953;
        double r751987 = r751986 * r751970;
        double r751988 = r751984 / r751987;
        double r751989 = r751950 ? r751983 : r751988;
        double r751990 = r751975 ? r751980 : r751989;
        double r751991 = r751948 ? r751973 : r751990;
        return r751991;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < 2.01827746078287e-310

    1. Initial program 16.8

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    2. Simplified16.8

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{2}}\\ \end{array}}\]
    3. Using strategy rm

    4. Applied flip3--16.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\color{blue}{\frac{{\left(-b\right)}^{3} - {\left(\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}^{3}}{\left(-b\right) \cdot \left(-b\right) + \left(\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} + \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{2}}\\ \end{array}\]

    5. Simplified16.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\frac{\color{blue}{-\mathsf{fma}\left(b, b \cdot b, \mathsf{fma}\left(c \cdot a, -4, b \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}\right)}}{\left(-b\right) \cdot \left(-b\right) + \left(\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} + \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{2}}\\ \end{array}\]

    6. Simplified16.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\frac{-\mathsf{fma}\left(b, b \cdot b, \mathsf{fma}\left(c \cdot a, -4, b \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b, \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}, b \cdot b\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{2}}\\ \end{array}\]

    if 2.01827746078287e-310 < b < 2.559678284282607e+69

    1. Initial program 9.2

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    2. Simplified9.2

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{2}}\\ \end{array}}\]

    3. Taylor expanded around 0 9.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{c}}{\frac{0}{2}}\\ \end{array}\]

    if 2.559678284282607e+69 < b

    1. Initial program 39.0

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    2. Simplified38.9

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{2}}\\ \end{array}}\]

    3. Taylor expanded around inf 9.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{2}}\\ \end{array}\]

    4. Simplified4.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{2}}\\ \end{array}\]

    5. Taylor expanded around 0 4.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\color{blue}{\frac{c}{b} - \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{2}}\\ \end{array}\]
    6. Using strategy rm

    7. Applied clear-num4.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{2}}{c}}\\ \end{array}\]
    8. Using strategy rm

    9. Applied *-un-lft-identity4.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{2}}{1 \cdot c}}}\\ \end{array}\]

    10. Applied div-inv4.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{1}}{\frac{\left(\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b\right) \cdot \frac{1}{2}}{1 \cdot c}}\\ \end{array}\]

    11. Applied times-frac4.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{1} \cdot \frac{\frac{1}{2}}{c}}}\\ \end{array}\]

    12. Simplified4.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{1}}{\left(\sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)} - b\right) \cdot \frac{\frac{1}{2}}{c}}\\ \end{array}\]

    13. Simplified4.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\color{blue}{\left(\sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)} - b\right) \cdot \frac{\frac{1}{2}}{c}}}\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification12.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2.01827746078287 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;-\frac{\frac{\mathsf{fma}\left(b, b \cdot b, \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}\right)}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} - b, \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}, b \cdot b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{\sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)} - b}{2}}\\ \end{array}\\ \mathbf{elif}\;b \le 2.559678284282607 \cdot 10^{+69}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{2}}{c} \cdot \left(\sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)} - b\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 1"
  (if (>= b 0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))))))