Average Error: 0.1 → 0.1
Time: 12.3s
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 r371910 = x;
        double r371911 = y;
        double r371912 = 0.5;
        double r371913 = r371911 + r371912;
        double r371914 = log(r371911);
        double r371915 = r371913 * r371914;
        double r371916 = r371910 - r371915;
        double r371917 = r371916 + r371911;
        double r371918 = z;
        double r371919 = r371917 - r371918;
        return r371919;
}
↓
double f(double x, double y, double z) {
        double r371920 = x;
        double r371921 = y;
        double r371922 = log(r371921);
        double r371923 = 0.5;
        double r371924 = r371921 + r371923;
        double r371925 = z;
        double r371926 = fma(r371922, r371924, r371925);
        double r371927 = r371926 - r371921;
        double r371928 = r371920 - r371927;
        return r371928;
}

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 2020001 +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))