Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Error: 0.1 → 0.1

Time: 4.6s

Precision: 64

Math

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

↓

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

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 (((x + y) + (-(0.5 * log(y)) - (log(y) * 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 sub-neg 0.1

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

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. Simplified 0.1

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

7. Applied distribute-lft-in 0.1

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

8. Applied associate--r+ 0.1

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

9. Simplified 0.1

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

11. Applied sub-neg 0.1

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

12. Applied associate--l+ 0.1

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

13. Applied associate-+r+ 0.1

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

14. Final simplification 0.1

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

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))