Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Error: 0.1 → 0.1

Time: 4.8s

Precision: 64

Math

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

↓

\[\left(x + \left(\left(\left(y - 0.5 \cdot \log y\right) - \log \left(\sqrt{y}\right) \cdot y\right) - \frac{y}{2} \cdot \log y\right)\right) - z\]

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(sqrt(y)) * y)) - ((y / 2.0) * log(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. Taylor expanded around 0 0.1

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

3. Simplified 0.1

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

5. Applied distribute-rgt-in 0.1

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

6. Applied associate--r+ 0.1

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

8. Applied add-sqr-sqrt 0.1

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

9. Applied log-prod 0.1

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

10. Applied distribute-lft-in 0.1

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

11. Applied associate--r+ 0.1

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

12. Simplified 0.1

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

14. Applied pow1/2 0.1

    \[\leadsto \left(x + \left(\left(\left(y - 0.5 \cdot \log y\right) - \log \left(\sqrt{y}\right) \cdot y\right) - y \cdot \log \color{blue}{\left({y}^{\frac{1}{2}}\right)}\right)\right) - z\]

15. Applied log-pow 0.1

    \[\leadsto \left(x + \left(\left(\left(y - 0.5 \cdot \log y\right) - \log \left(\sqrt{y}\right) \cdot y\right) - y \cdot \color{blue}{\left(\frac{1}{2} \cdot \log y\right)}\right)\right) - z\]

16. Applied associate-*r* 0.1

    \[\leadsto \left(x + \left(\left(\left(y - 0.5 \cdot \log y\right) - \log \left(\sqrt{y}\right) \cdot y\right) - \color{blue}{\left(y \cdot \frac{1}{2}\right) \cdot \log y}\right)\right) - z\]

17. Simplified 0.1

    \[\leadsto \left(x + \left(\left(\left(y - 0.5 \cdot \log y\right) - \log \left(\sqrt{y}\right) \cdot y\right) - \color{blue}{\frac{y}{2}} \cdot \log y\right)\right) - z\]

18. Final simplification 0.1

    \[\leadsto \left(x + \left(\left(\left(y - 0.5 \cdot \log y\right) - \log \left(\sqrt{y}\right) \cdot y\right) - \frac{y}{2} \cdot \log y\right)\right) - z\]

Reproduce

herbie shell --seed 2020075 
(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))