Average Error: 43.4 → 11.4
Time: 12.0s
Precision: 64
\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.001614548699194859283576053421427332068561:\\ \;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 0.001614548699194859283576053421427332068561:\\
\;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r29942 = b;
        double r29943 = -r29942;
        double r29944 = r29942 * r29942;
        double r29945 = 4.0;
        double r29946 = a;
        double r29947 = r29945 * r29946;
        double r29948 = c;
        double r29949 = r29947 * r29948;
        double r29950 = r29944 - r29949;
        double r29951 = sqrt(r29950);
        double r29952 = r29943 + r29951;
        double r29953 = 2.0;
        double r29954 = r29953 * r29946;
        double r29955 = r29952 / r29954;
        return r29955;
}
↓
double f(double a, double b, double c) {
        double r29956 = b;
        double r29957 = 0.0016145486991948593;
        bool r29958 = r29956 <= r29957;
        double r29959 = r29956 * r29956;
        double r29960 = 4.0;
        double r29961 = a;
        double r29962 = r29960 * r29961;
        double r29963 = c;
        double r29964 = r29962 * r29963;
        double r29965 = fma(r29956, r29956, r29964);
        double r29966 = r29959 - r29965;
        double r29967 = r29959 - r29964;
        double r29968 = sqrt(r29967);
        double r29969 = r29968 + r29956;
        double r29970 = r29966 / r29969;
        double r29971 = 2.0;
        double r29972 = r29971 * r29961;
        double r29973 = r29970 / r29972;
        double r29974 = -1.0;
        double r29975 = r29963 / r29956;
        double r29976 = r29974 * r29975;
        double r29977 = r29958 ? r29973 : r29976;
        return r29977;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 0.0016145486991948593

    1. Initial program 20.0

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

    2. Simplified20.0

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

    4. Applied flip--20.0

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

    5. Simplified19.2

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

    if 0.0016145486991948593 < b

    1. Initial program 45.8

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

    2. Simplified45.8

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

    3. Taylor expanded around inf 10.6

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.001614548699194859283576053421427332068561:\\ \;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e15) (< 1.11022e-16 b 9.0072e15) (< 1.11022e-16 c 9.0072e15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))