Average Error: 7.0 → 0.3
Time: 9.6s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(\log y, x - 1, \mathsf{fma}\left(z - 1, \log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right), -t\right)\right)\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\mathsf{fma}\left(\log y, x - 1, \mathsf{fma}\left(z - 1, \log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right), -t\right)\right)
double f(double x, double y, double z, double t) {
        double r61077 = x;
        double r61078 = 1.0;
        double r61079 = r61077 - r61078;
        double r61080 = y;
        double r61081 = log(r61080);
        double r61082 = r61079 * r61081;
        double r61083 = z;
        double r61084 = r61083 - r61078;
        double r61085 = r61078 - r61080;
        double r61086 = log(r61085);
        double r61087 = r61084 * r61086;
        double r61088 = r61082 + r61087;
        double r61089 = t;
        double r61090 = r61088 - r61089;
        return r61090;
}
↓
double f(double x, double y, double z, double t) {
        double r61091 = y;
        double r61092 = log(r61091);
        double r61093 = x;
        double r61094 = 1.0;
        double r61095 = r61093 - r61094;
        double r61096 = z;
        double r61097 = r61096 - r61094;
        double r61098 = log(r61094);
        double r61099 = r61094 * r61091;
        double r61100 = 0.5;
        double r61101 = 2.0;
        double r61102 = pow(r61091, r61101);
        double r61103 = pow(r61094, r61101);
        double r61104 = r61102 / r61103;
        double r61105 = r61100 * r61104;
        double r61106 = r61099 + r61105;
        double r61107 = r61098 - r61106;
        double r61108 = t;
        double r61109 = -r61108;
        double r61110 = fma(r61097, r61107, r61109);
        double r61111 = fma(r61092, r61095, r61110);
        return r61111;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 7.0

    \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]

  2. Simplified7.0

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

  3. Taylor expanded around 0 0.3

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

  5. Applied fma-neg0.3

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

  6. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(\log y, x - 1, \mathsf{fma}\left(z - 1, \log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right), -t\right)\right)\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  (- (+ (* (- x 1) (log y)) (* (- z 1) (log (- 1 y)))) t))