Average Error: 33.2 → 9.7
Time: 26.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.170773079316174 \cdot 10^{+99}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 3.0168583404714427 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -4.170773079316174 \cdot 10^{+99}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 3.0168583404714427 \cdot 10^{-66}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}{a} - \frac{b}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r1549943 = b;
        double r1549944 = -r1549943;
        double r1549945 = r1549943 * r1549943;
        double r1549946 = 4.0;
        double r1549947 = a;
        double r1549948 = r1549946 * r1549947;
        double r1549949 = c;
        double r1549950 = r1549948 * r1549949;
        double r1549951 = r1549945 - r1549950;
        double r1549952 = sqrt(r1549951);
        double r1549953 = r1549944 + r1549952;
        double r1549954 = 2.0;
        double r1549955 = r1549954 * r1549947;
        double r1549956 = r1549953 / r1549955;
        return r1549956;
}
↓
double f(double a, double b, double c) {
        double r1549957 = b;
        double r1549958 = -4.170773079316174e+99;
        bool r1549959 = r1549957 <= r1549958;
        double r1549960 = c;
        double r1549961 = r1549960 / r1549957;
        double r1549962 = a;
        double r1549963 = r1549957 / r1549962;
        double r1549964 = r1549961 - r1549963;
        double r1549965 = 2.0;
        double r1549966 = r1549964 * r1549965;
        double r1549967 = r1549966 / r1549965;
        double r1549968 = 3.0168583404714427e-66;
        bool r1549969 = r1549957 <= r1549968;
        double r1549970 = r1549962 * r1549960;
        double r1549971 = -4.0;
        double r1549972 = r1549970 * r1549971;
        double r1549973 = fma(r1549957, r1549957, r1549972);
        double r1549974 = sqrt(r1549973);
        double r1549975 = r1549974 / r1549962;
        double r1549976 = r1549975 - r1549963;
        double r1549977 = r1549976 / r1549965;
        double r1549978 = -2.0;
        double r1549979 = r1549961 * r1549978;
        double r1549980 = r1549979 / r1549965;
        double r1549981 = r1549969 ? r1549977 : r1549980;
        double r1549982 = r1549959 ? r1549967 : r1549981;
        return r1549982;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -4.170773079316174e+99

    1. Initial program 44.2

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

    2. Simplified44.2

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

    3. Taylor expanded around 0 44.2

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

    5. Applied div-sub44.2

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

    6. Taylor expanded around -inf 3.3

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

    7. Simplified3.3

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

    if -4.170773079316174e+99 < b < 3.0168583404714427e-66

    1. Initial program 12.8

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

    2. Simplified12.8

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

    3. Taylor expanded around 0 12.8

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

    5. Applied div-sub12.8

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

    if 3.0168583404714427e-66 < b

    1. Initial program 53.1

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

    2. Simplified53.1

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

    3. Taylor expanded around 0 53.0

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

    5. Applied div-sub53.7

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

    6. Taylor expanded around inf 8.5

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.170773079316174 \cdot 10^{+99}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 3.0168583404714427 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))