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

Average Error: 0.1 → 0.1

Time: 16.7s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

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

↓

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

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

4. Applied log-prod 0.1

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

5. Applied distribute-lft-in 0.1

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

6. Applied associate-+r+ 0.1

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

7. Simplified 0.1

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

9. Applied add-sqr-sqrt 0.1

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

10. Applied sqrt-prod 0.1

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

11. Applied log-prod 0.1

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

12. Applied distribute-lft-in 0.1

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

13. Applied associate-+r+ 0.1

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

14. Simplified 0.1

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

15. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020182 
(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)))