Average Error: 0.0 → 0.6
Time: 6.7s
Precision: 64
\[\left(\left(x - \left(y - 1\right) \cdot z\right) - \left(t - 1\right) \cdot a\right) + \left(\left(y + t\right) - 2\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7838888393858995980000 \lor \neg \left(t \le 3.2768003103463893 \cdot 10^{-44}\right):\\ \;\;\;\;\mathsf{fma}\left(1 - y, z, \mathsf{fma}\left(b, y, \mathsf{fma}\left(b, t, x\right)\right) - \left(t - 1\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - y, z, \mathsf{fma}\left(b, \left(y + t\right) - 2, x\right) - \frac{\left(t \cdot t - 1 \cdot 1\right) \cdot a}{t + 1}\right)\\ \end{array}\]
\left(\left(x - \left(y - 1\right) \cdot z\right) - \left(t - 1\right) \cdot a\right) + \left(\left(y + t\right) - 2\right) \cdot b
↓
\begin{array}{l}
\mathbf{if}\;t \le -7838888393858995980000 \lor \neg \left(t \le 3.2768003103463893 \cdot 10^{-44}\right):\\
\;\;\;\;\mathsf{fma}\left(1 - y, z, \mathsf{fma}\left(b, y, \mathsf{fma}\left(b, t, x\right)\right) - \left(t - 1\right) \cdot a\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(1 - y, z, \mathsf{fma}\left(b, \left(y + t\right) - 2, x\right) - \frac{\left(t \cdot t - 1 \cdot 1\right) \cdot a}{t + 1}\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r50983 = x;
        double r50984 = y;
        double r50985 = 1.0;
        double r50986 = r50984 - r50985;
        double r50987 = z;
        double r50988 = r50986 * r50987;
        double r50989 = r50983 - r50988;
        double r50990 = t;
        double r50991 = r50990 - r50985;
        double r50992 = a;
        double r50993 = r50991 * r50992;
        double r50994 = r50989 - r50993;
        double r50995 = r50984 + r50990;
        double r50996 = 2.0;
        double r50997 = r50995 - r50996;
        double r50998 = b;
        double r50999 = r50997 * r50998;
        double r51000 = r50994 + r50999;
        return r51000;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r51001 = t;
        double r51002 = -7.838888393858996e+21;
        bool r51003 = r51001 <= r51002;
        double r51004 = 3.2768003103463893e-44;
        bool r51005 = r51001 <= r51004;
        double r51006 = !r51005;
        bool r51007 = r51003 || r51006;
        double r51008 = 1.0;
        double r51009 = y;
        double r51010 = r51008 - r51009;
        double r51011 = z;
        double r51012 = b;
        double r51013 = x;
        double r51014 = fma(r51012, r51001, r51013);
        double r51015 = fma(r51012, r51009, r51014);
        double r51016 = r51001 - r51008;
        double r51017 = a;
        double r51018 = r51016 * r51017;
        double r51019 = r51015 - r51018;
        double r51020 = fma(r51010, r51011, r51019);
        double r51021 = r51009 + r51001;
        double r51022 = 2.0;
        double r51023 = r51021 - r51022;
        double r51024 = fma(r51012, r51023, r51013);
        double r51025 = r51001 * r51001;
        double r51026 = r51008 * r51008;
        double r51027 = r51025 - r51026;
        double r51028 = r51027 * r51017;
        double r51029 = r51001 + r51008;
        double r51030 = r51028 / r51029;
        double r51031 = r51024 - r51030;
        double r51032 = fma(r51010, r51011, r51031);
        double r51033 = r51007 ? r51020 : r51032;
        return r51033;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Derivation

  1. Split input into 2 regimes

  2. if t < -7.838888393858996e+21 or 3.2768003103463893e-44 < t

    1. Initial program 0.0

      \[\left(\left(x - \left(y - 1\right) \cdot z\right) - \left(t - 1\right) \cdot a\right) + \left(\left(y + t\right) - 2\right) \cdot b\]

    2. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - y, z, \mathsf{fma}\left(b, \left(y + t\right) - 2, x\right) - \left(t - 1\right) \cdot a\right)}\]

    3. Taylor expanded around inf 1.3

      \[\leadsto \mathsf{fma}\left(1 - y, z, \color{blue}{\left(y \cdot b + \left(x + t \cdot b\right)\right)} - \left(t - 1\right) \cdot a\right)\]

    4. Simplified1.3

      \[\leadsto \mathsf{fma}\left(1 - y, z, \color{blue}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(b, t, x\right)\right)} - \left(t - 1\right) \cdot a\right)\]

    if -7.838888393858996e+21 < t < 3.2768003103463893e-44

    1. Initial program 0.0

      \[\left(\left(x - \left(y - 1\right) \cdot z\right) - \left(t - 1\right) \cdot a\right) + \left(\left(y + t\right) - 2\right) \cdot b\]

    2. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - y, z, \mathsf{fma}\left(b, \left(y + t\right) - 2, x\right) - \left(t - 1\right) \cdot a\right)}\]
    3. Using strategy rm

    4. Applied flip--0.0

      \[\leadsto \mathsf{fma}\left(1 - y, z, \mathsf{fma}\left(b, \left(y + t\right) - 2, x\right) - \color{blue}{\frac{t \cdot t - 1 \cdot 1}{t + 1}} \cdot a\right)\]

    5. Applied associate-*l/0.1

      \[\leadsto \mathsf{fma}\left(1 - y, z, \mathsf{fma}\left(b, \left(y + t\right) - 2, x\right) - \color{blue}{\frac{\left(t \cdot t - 1 \cdot 1\right) \cdot a}{t + 1}}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -7838888393858995980000 \lor \neg \left(t \le 3.2768003103463893 \cdot 10^{-44}\right):\\ \;\;\;\;\mathsf{fma}\left(1 - y, z, \mathsf{fma}\left(b, y, \mathsf{fma}\left(b, t, x\right)\right) - \left(t - 1\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - y, z, \mathsf{fma}\left(b, \left(y + t\right) - 2, x\right) - \frac{\left(t \cdot t - 1 \cdot 1\right) \cdot a}{t + 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (x y z t a b)
  :name "Statistics.Distribution.Beta:$centropy from math-functions-0.1.5.2"
  :precision binary64
  (+ (- (- x (* (- y 1) z)) (* (- t 1) a)) (* (- (+ y t) 2) b)))