Average Error: 0.1 → 0.1
Time: 15.8s
Precision: 64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
↓
\[\mathsf{fma}\left(0.5 + y, -\log y, y + x\right) - z\]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
\mathsf{fma}\left(0.5 + y, -\log y, y + x\right) - z
double f(double x, double y, double z) {
        double r232853 = x;
        double r232854 = y;
        double r232855 = 0.5;
        double r232856 = r232854 + r232855;
        double r232857 = log(r232854);
        double r232858 = r232856 * r232857;
        double r232859 = r232853 - r232858;
        double r232860 = r232859 + r232854;
        double r232861 = z;
        double r232862 = r232860 - r232861;
        return r232862;
}
↓
double f(double x, double y, double z) {
        double r232863 = 0.5;
        double r232864 = y;
        double r232865 = r232863 + r232864;
        double r232866 = log(r232864);
        double r232867 = -r232866;
        double r232868 = x;
        double r232869 = r232864 + r232868;
        double r232870 = fma(r232865, r232867, r232869);
        double r232871 = z;
        double r232872 = r232870 - r232871;
        return r232872;
}

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. Using strategy rm

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

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

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

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

  5. Applied distribute-lft-out0.1

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

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"

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

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