Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, B

Average Error: 0.1 → 0.1

Time: 9.9s

Precision: 64

Math

\[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

↓

\[\left(\left(\left(\left(x \cdot \log \left(\sqrt{\sqrt{y}}\right) + \left(z + x \cdot \left(\log \left(\sqrt{\sqrt{y}}\right) + \log \left(\sqrt{y}\right)\right)\right)\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return ((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i));
}

↓

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return ((((((x * log(sqrt(sqrt(y)))) + (z + (x * (log(sqrt(sqrt(y))) + log(sqrt(y)))))) + t) + a) + ((b - 0.5) * log(c))) + (y * i));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus i

Derivation

1. Initial program 0.1

   \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]
2. Using strategy rm

3. Applied add-sqr-sqrt 0.1

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

4. Applied log-prod 0.1

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

5. Applied distribute-lft-in 0.1

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

6. Applied associate-+l+ 0.1

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

7. Simplified 0.1

   \[\leadsto \left(\left(\left(\left(x \cdot \log \left(\sqrt{y}\right) + \color{blue}{\left(\log \left(\sqrt{y}\right) \cdot x + z\right)}\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]
8. Using strategy rm

9. Applied add-sqr-sqrt 0.1

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

10. Applied sqrt-prod 0.1

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

11. Applied log-prod 0.1

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

12. Applied distribute-lft-in 0.1

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

13. Applied associate-+l+ 0.1

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

14. Simplified 0.1

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

15. Final simplification 0.1

    \[\leadsto \left(\left(\left(\left(x \cdot \log \left(\sqrt{\sqrt{y}}\right) + \left(z + x \cdot \left(\log \left(\sqrt{\sqrt{y}}\right) + \log \left(\sqrt{y}\right)\right)\right)\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

Reproduce

herbie shell --seed 2020106 
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, B"
  :precision binary64
  (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))