Average Error: 0.1 → 0.1
Time: 19.6s
Precision: 64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
↓
\[\left(x + \left(\left(\log y - 1\right) \cdot \left(-y\right) - \log y \cdot 0.5\right)\right) - z\]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
\left(x + \left(\left(\log y - 1\right) \cdot \left(-y\right) - \log y \cdot 0.5\right)\right) - z
double f(double x, double y, double z) {
        double r18908993 = x;
        double r18908994 = y;
        double r18908995 = 0.5;
        double r18908996 = r18908994 + r18908995;
        double r18908997 = log(r18908994);
        double r18908998 = r18908996 * r18908997;
        double r18908999 = r18908993 - r18908998;
        double r18909000 = r18908999 + r18908994;
        double r18909001 = z;
        double r18909002 = r18909000 - r18909001;
        return r18909002;
}
↓
double f(double x, double y, double z) {
        double r18909003 = x;
        double r18909004 = y;
        double r18909005 = log(r18909004);
        double r18909006 = 1.0;
        double r18909007 = r18909005 - r18909006;
        double r18909008 = -r18909004;
        double r18909009 = r18909007 * r18909008;
        double r18909010 = 0.5;
        double r18909011 = r18909005 * r18909010;
        double r18909012 = r18909009 - r18909011;
        double r18909013 = r18909003 + r18909012;
        double r18909014 = z;
        double r18909015 = r18909013 - r18909014;
        return r18909015;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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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

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

  7. Applied distribute-rgt-in0.1

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

  8. Applied associate--r+0.1

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

  9. Taylor expanded around -inf 62.5

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

  10. Simplified0.1

    \[\leadsto \left(x + \left(\color{blue}{\left(-y\right) \cdot \left(\left(0 + \log y\right) - 1\right)} - 0.5 \cdot \log y\right)\right) - z\]

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019163 
(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))