Average Error: 0.1 → 0.1
Time: 6.4s
Precision: 64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
↓
\[\left(x + \mathsf{fma}\left(-\left(y + 0.5\right), \log y, y\right)\right) - z\]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
\left(x + \mathsf{fma}\left(-\left(y + 0.5\right), \log y, y\right)\right) - z
double f(double x, double y, double z) {
        double r328979 = x;
        double r328980 = y;
        double r328981 = 0.5;
        double r328982 = r328980 + r328981;
        double r328983 = log(r328980);
        double r328984 = r328982 * r328983;
        double r328985 = r328979 - r328984;
        double r328986 = r328985 + r328980;
        double r328987 = z;
        double r328988 = r328986 - r328987;
        return r328988;
}
↓
double f(double x, double y, double z) {
        double r328989 = x;
        double r328990 = y;
        double r328991 = 0.5;
        double r328992 = r328990 + r328991;
        double r328993 = -r328992;
        double r328994 = log(r328990);
        double r328995 = fma(r328993, r328994, r328990);
        double r328996 = r328989 + r328995;
        double r328997 = z;
        double r328998 = r328996 - r328997;
        return r328998;
}

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 sub-neg0.1

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

  4. Applied associate-+l+0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2020035 +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))