Average Error: 9.5 → 0.3
Time: 9.7s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(\log y, x, z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right) - t\right)\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\mathsf{fma}\left(\log y, x, z \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right) - t\right)
double f(double x, double y, double z, double t) {
        double r418862 = x;
        double r418863 = y;
        double r418864 = log(r418863);
        double r418865 = r418862 * r418864;
        double r418866 = z;
        double r418867 = 1.0;
        double r418868 = r418867 - r418863;
        double r418869 = log(r418868);
        double r418870 = r418866 * r418869;
        double r418871 = r418865 + r418870;
        double r418872 = t;
        double r418873 = r418871 - r418872;
        return r418873;
}
↓
double f(double x, double y, double z, double t) {
        double r418874 = y;
        double r418875 = log(r418874);
        double r418876 = x;
        double r418877 = z;
        double r418878 = 1.0;
        double r418879 = log(r418878);
        double r418880 = r418878 * r418874;
        double r418881 = 0.5;
        double r418882 = 2.0;
        double r418883 = pow(r418874, r418882);
        double r418884 = pow(r418878, r418882);
        double r418885 = r418883 / r418884;
        double r418886 = r418881 * r418885;
        double r418887 = r418880 + r418886;
        double r418888 = r418879 - r418887;
        double r418889 = r418877 * r418888;
        double r418890 = t;
        double r418891 = r418889 - r418890;
        double r418892 = fma(r418875, r418876, r418891);
        return r418892;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.5
Target0.3
Herbie0.3
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.333333333333333315}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Initial program 9.5

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

  2. Simplified9.5

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

  3. Taylor expanded around 0 0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2020081 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (- (* (- z) (+ (+ (* 0.5 (* y y)) y) (* (/ 0.3333333333333333 (* 1 (* 1 1))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1 y)))) t))