Average Error: 0.1 → 0.1
Time: 18.1s
Precision: 64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
↓
\[\left(x + \mathsf{fma}\left(-\log y, y + 0.5, y\right)\right) - z\]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
\left(x + \mathsf{fma}\left(-\log y, y + 0.5, y\right)\right) - z
double f(double x, double y, double z) {
        double r282861 = x;
        double r282862 = y;
        double r282863 = 0.5;
        double r282864 = r282862 + r282863;
        double r282865 = log(r282862);
        double r282866 = r282864 * r282865;
        double r282867 = r282861 - r282866;
        double r282868 = r282867 + r282862;
        double r282869 = z;
        double r282870 = r282868 - r282869;
        return r282870;
}
↓
double f(double x, double y, double z) {
        double r282871 = x;
        double r282872 = y;
        double r282873 = log(r282872);
        double r282874 = -r282873;
        double r282875 = 0.5;
        double r282876 = r282872 + r282875;
        double r282877 = fma(r282874, r282876, r282872);
        double r282878 = r282871 + r282877;
        double r282879 = z;
        double r282880 = r282878 - r282879;
        return r282880;
}

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

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019212 +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))