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

Average Error: 0.1 → 0.1

Time: 7.5s

Precision: binary64

Math

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

↓

\[\left(\left(\left(\left(\left(x \cdot \left(\log \left(\sqrt[3]{y}\right) \cdot 2 + \log \left(\sqrt[3]{\sqrt{y}}\right)\right) + x \cdot \log \left(\sqrt[3]{\sqrt{y}}\right)\right) + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))

↓

(FPCore (x y z t a b c i)
 :precision binary64
 (+
  (+
   (+
    (+
     (+
      (+
       (* x (+ (* (log (cbrt y)) 2.0) (log (cbrt (sqrt y)))))
       (* x (log (cbrt (sqrt y)))))
      z)
     t)
    a)
   (* (- b 0.5) (log c)))
  (* y 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) (((double) (((double) (x * ((double) (((double) (((double) log(((double) cbrt(y)))) * 2.0)) + ((double) log(((double) cbrt(((double) sqrt(y)))))))))) + ((double) (x * ((double) log(((double) cbrt(((double) sqrt(y)))))))))) + z)) + t)) + a)) + ((double) (((double) (b - 0.5)) * ((double) log(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 Error: 0.1 bits

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

3. Applied add-cube-cbrt Error: 0.1 bits

   \[\leadsto \left(\left(\left(\left(x \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

4. Applied log-prod Error: 0.1 bits

   \[\leadsto \left(\left(\left(\left(x \cdot \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

5. Applied distribute-lft-in Error: 0.1 bits

   \[\leadsto \left(\left(\left(\left(\color{blue}{\left(x \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right)} + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

6. Simplified Error: 0.1 bits

   \[\leadsto \left(\left(\left(\left(\left(\color{blue}{x \cdot \left(\log \left(\sqrt[3]{y}\right) \cdot 2\right)} + x \cdot \log \left(\sqrt[3]{y}\right)\right) + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]
7. Using strategy rm

8. Applied add-sqr-sqrt Error: 0.1 bits

   \[\leadsto \left(\left(\left(\left(\left(x \cdot \left(\log \left(\sqrt[3]{y}\right) \cdot 2\right) + x \cdot \log \left(\sqrt[3]{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}\right)\right) + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

9. Applied cbrt-prod Error: 0.1 bits

   \[\leadsto \left(\left(\left(\left(\left(x \cdot \left(\log \left(\sqrt[3]{y}\right) \cdot 2\right) + x \cdot \log \color{blue}{\left(\sqrt[3]{\sqrt{y}} \cdot \sqrt[3]{\sqrt{y}}\right)}\right) + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

10. Applied log-prod Error: 0.1 bits

    \[\leadsto \left(\left(\left(\left(\left(x \cdot \left(\log \left(\sqrt[3]{y}\right) \cdot 2\right) + x \cdot \color{blue}{\left(\log \left(\sqrt[3]{\sqrt{y}}\right) + \log \left(\sqrt[3]{\sqrt{y}}\right)\right)}\right) + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

11. Applied distribute-lft-in Error: 0.1 bits

    \[\leadsto \left(\left(\left(\left(\left(x \cdot \left(\log \left(\sqrt[3]{y}\right) \cdot 2\right) + \color{blue}{\left(x \cdot \log \left(\sqrt[3]{\sqrt{y}}\right) + x \cdot \log \left(\sqrt[3]{\sqrt{y}}\right)\right)}\right) + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

12. Applied associate-+r+ Error: 0.1 bits

    \[\leadsto \left(\left(\left(\left(\color{blue}{\left(\left(x \cdot \left(\log \left(\sqrt[3]{y}\right) \cdot 2\right) + x \cdot \log \left(\sqrt[3]{\sqrt{y}}\right)\right) + x \cdot \log \left(\sqrt[3]{\sqrt{y}}\right)\right)} + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

13. Simplified Error: 0.1 bits

    \[\leadsto \left(\left(\left(\left(\left(\color{blue}{x \cdot \left(\log \left(\sqrt[3]{y}\right) \cdot 2 + \log \left(\sqrt[3]{\sqrt{y}}\right)\right)} + x \cdot \log \left(\sqrt[3]{\sqrt{y}}\right)\right) + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

14. Final simplification Error: 0.1 bits

    \[\leadsto \left(\left(\left(\left(\left(x \cdot \left(\log \left(\sqrt[3]{y}\right) \cdot 2 + \log \left(\sqrt[3]{\sqrt{y}}\right)\right) + x \cdot \log \left(\sqrt[3]{\sqrt{y}}\right)\right) + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

Reproduce

herbie shell --seed 2020205 
(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)))