Average Error: 7.0 → 0.3
Time: 9.7s
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 r61069 = x;
        double r61070 = 1.0;
        double r61071 = r61069 - r61070;
        double r61072 = y;
        double r61073 = log(r61072);
        double r61074 = r61071 * r61073;
        double r61075 = z;
        double r61076 = r61075 - r61070;
        double r61077 = r61070 - r61072;
        double r61078 = log(r61077);
        double r61079 = r61076 * r61078;
        double r61080 = r61074 + r61079;
        double r61081 = t;
        double r61082 = r61080 - r61081;
        return r61082;
}
↓
double f(double x, double y, double z, double t) {
        double r61083 = y;
        double r61084 = log(r61083);
        double r61085 = x;
        double r61086 = 1.0;
        double r61087 = r61085 - r61086;
        double r61088 = z;
        double r61089 = r61088 - r61086;
        double r61090 = log(r61086);
        double r61091 = r61086 * r61083;
        double r61092 = 0.5;
        double r61093 = 2.0;
        double r61094 = pow(r61083, r61093);
        double r61095 = pow(r61086, r61093);
        double r61096 = r61094 / r61095;
        double r61097 = r61092 * r61096;
        double r61098 = r61091 + r61097;
        double r61099 = r61090 - r61098;
        double r61100 = t;
        double r61101 = -r61100;
        double r61102 = fma(r61089, r61099, r61101);
        double r61103 = fma(r61084, r61087, r61102);
        return r61103;
}

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))