Average Error: 0.1 → 0.1
Time: 4.2s
Precision: binary64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
↓
\[x + \left(y - \left(\left(\left(y + 0.5\right) \cdot \log \left(\sqrt{y}\right) + \left(y + 0.5\right) \cdot \log \left(\sqrt{y}\right)\right) + z\right)\right)\]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
x + \left(y - \left(\left(\left(y + 0.5\right) \cdot \log \left(\sqrt{y}\right) + \left(y + 0.5\right) \cdot \log \left(\sqrt{y}\right)\right) + z\right)\right)
double code(double x, double y, double z) {
	return ((double) (((double) (((double) (x - ((double) (((double) (y + 0.5)) * ((double) log(y)))))) + y)) - z));
}
↓
double code(double x, double y, double z) {
	return ((double) (x + ((double) (y - ((double) (((double) (((double) (((double) (y + 0.5)) * ((double) log(((double) sqrt(y)))))) + ((double) (((double) (y + 0.5)) * ((double) log(((double) sqrt(y)))))))) + z))))));
}

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. Simplified0.1

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

  4. Applied add-sqr-sqrt0.1

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

  5. Applied log-prod0.1

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

  6. Applied distribute-lft-in0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2020179 
(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))