Average Error: 0.1 → 0.1
Time: 9.0s
Precision: 64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
↓
\[x - \left(\mathsf{fma}\left(\log y, y + 0.5, z\right) - y\right)\]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
x - \left(\mathsf{fma}\left(\log y, y + 0.5, z\right) - y\right)
double f(double x, double y, double z) {
        double r322881 = x;
        double r322882 = y;
        double r322883 = 0.5;
        double r322884 = r322882 + r322883;
        double r322885 = log(r322882);
        double r322886 = r322884 * r322885;
        double r322887 = r322881 - r322886;
        double r322888 = r322887 + r322882;
        double r322889 = z;
        double r322890 = r322888 - r322889;
        return r322890;
}
↓
double f(double x, double y, double z) {
        double r322891 = x;
        double r322892 = y;
        double r322893 = log(r322892);
        double r322894 = 0.5;
        double r322895 = r322892 + r322894;
        double r322896 = z;
        double r322897 = fma(r322893, r322895, r322896);
        double r322898 = r322897 - r322892;
        double r322899 = r322891 - r322898;
        return r322899;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.1
Target0.1
Herbie0.1
\[\left(\left(y + x\right) - z\right) - \left(y + 0.5\right) \cdot \log y\]

Derivation

  1. Initial program 0.1

    \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]

  2. Simplified0.1

    \[\leadsto \color{blue}{x - \left(\mathsf{fma}\left(\log y, y + 0.5, z\right) - y\right)}\]

  3. Final simplification0.1

    \[\leadsto x - \left(\mathsf{fma}\left(\log y, y + 0.5, z\right) - y\right)\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (- (- (+ y x) z) (* (+ y 0.5) (log y)))

  (- (+ (- x (* (+ y 0.5) (log y))) y) z))