Average Error: 0.1 → 0.1
Time: 8.7s
Precision: 64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
↓
\[x - 1 \cdot \left(\mathsf{fma}\left(\log y, y + 0.5, z\right) - y\right)\]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
x - 1 \cdot \left(\mathsf{fma}\left(\log y, y + 0.5, z\right) - y\right)
double f(double x, double y, double z) {
        double r309916 = x;
        double r309917 = y;
        double r309918 = 0.5;
        double r309919 = r309917 + r309918;
        double r309920 = log(r309917);
        double r309921 = r309919 * r309920;
        double r309922 = r309916 - r309921;
        double r309923 = r309922 + r309917;
        double r309924 = z;
        double r309925 = r309923 - r309924;
        return r309925;
}
↓
double f(double x, double y, double z) {
        double r309926 = x;
        double r309927 = 1.0;
        double r309928 = y;
        double r309929 = log(r309928);
        double r309930 = 0.5;
        double r309931 = r309928 + r309930;
        double r309932 = z;
        double r309933 = fma(r309929, r309931, r309932);
        double r309934 = r309933 - r309928;
        double r309935 = r309927 * r309934;
        double r309936 = r309926 - r309935;
        return r309936;
}

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. Using strategy rm

  4. Applied *-un-lft-identity0.1

    \[\leadsto x - \color{blue}{1 \cdot \left(\mathsf{fma}\left(\log y, y + 0.5, z\right) - y\right)}\]

  5. Final simplification0.1

    \[\leadsto x - 1 \cdot \left(\mathsf{fma}\left(\log y, y + 0.5, z\right) - y\right)\]

Reproduce

herbie shell --seed 2020036 +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))