Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Error: 0.1 → 0.1

Time: 4.8s

Precision: 64

Math

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

↓

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

C

double code(double x, double y, double z) {
	return (((x - ((y + 0.5) * log(y))) + y) - z);
}

↓

double code(double x, double y, double z) {
	return ((1.0 * (x + (y - (log(y) * (0.5 + y))))) - z);
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	0.1
Target	0.1
Herbie	0.1

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

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-identity 0.1

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

4. Applied *-un-lft-identity 0.1

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

5. Applied distribute-lft-out 0.1

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

6. Simplified 0.1

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

7. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020078 
(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))