Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 5.7s

Precision: binary64

Math

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

↓

\[\log \left(x + y\right) + \left(\log z + \left(\left(\left(a - 0.5\right) \cdot \left(\log t \cdot 0.8333333333333334\right) + \left(a - 0.5\right) \cdot \log \left({t}^{0.16666666666666666}\right)\right) - t\right)\right)\]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (+
  (log (+ x y))
  (+
   (log z)
   (-
    (+
     (* (- a 0.5) (* (log t) 0.8333333333333334))
     (* (- a 0.5) (log (pow t 0.16666666666666666))))
    t))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	return ((double) (((double) log(((double) (x + y)))) + ((double) (((double) log(z)) + ((double) (((double) (((double) (((double) (a - 0.5)) * ((double) (((double) log(t)) * 0.8333333333333334)))) + ((double) (((double) (a - 0.5)) * ((double) log(((double) pow(t, 0.16666666666666666)))))))) - t))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	0.3
Target	0.3
Herbie	0.3

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

Derivation

1. Initial program Error: 0.3 bits

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

2. Simplified Error: 0.3 bits

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

4. Applied add-cube-cbrt Error: 0.3 bits

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

5. Applied log-prod Error: 0.3 bits

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

6. Applied distribute-lft-in Error: 0.3 bits

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

7. Simplified Error: 0.3 bits

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

9. Applied pow1/3 Error: 0.3 bits

   \[\leadsto \log \left(x + y\right) + \left(\log z + \left(\left(\left(a - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{t}\right)\right) + \left(a - 0.5\right) \cdot \log \color{blue}{\left({t}^{0.3333333333333333}\right)}\right) - t\right)\right)\]
10. Using strategy rm

11. Applied sqr-pow Error: 0.3 bits

    \[\leadsto \log \left(x + y\right) + \left(\log z + \left(\left(\left(a - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{t}\right)\right) + \left(a - 0.5\right) \cdot \log \color{blue}{\left({t}^{\left(\frac{0.3333333333333333}{2}\right)} \cdot {t}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}\right) - t\right)\right)\]

12. Applied log-prod Error: 0.3 bits

    \[\leadsto \log \left(x + y\right) + \left(\log z + \left(\left(\left(a - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{t}\right)\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(\log \left({t}^{\left(\frac{0.3333333333333333}{2}\right)}\right) + \log \left({t}^{\left(\frac{0.3333333333333333}{2}\right)}\right)\right)}\right) - t\right)\right)\]

13. Applied distribute-lft-in Error: 0.3 bits

    \[\leadsto \log \left(x + y\right) + \left(\log z + \left(\left(\left(a - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{t}\right)\right) + \color{blue}{\left(\left(a - 0.5\right) \cdot \log \left({t}^{\left(\frac{0.3333333333333333}{2}\right)}\right) + \left(a - 0.5\right) \cdot \log \left({t}^{\left(\frac{0.3333333333333333}{2}\right)}\right)\right)}\right) - t\right)\right)\]

14. Applied associate-+r+ Error: 0.3 bits

    \[\leadsto \log \left(x + y\right) + \left(\log z + \left(\color{blue}{\left(\left(\left(a - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{t}\right)\right) + \left(a - 0.5\right) \cdot \log \left({t}^{\left(\frac{0.3333333333333333}{2}\right)}\right)\right) + \left(a - 0.5\right) \cdot \log \left({t}^{\left(\frac{0.3333333333333333}{2}\right)}\right)\right)} - t\right)\right)\]

15. Simplified Error: 0.3 bits

    \[\leadsto \log \left(x + y\right) + \left(\log z + \left(\left(\color{blue}{\left(a - 0.5\right) \cdot \left(\log t \cdot 0.8333333333333334\right)} + \left(a - 0.5\right) \cdot \log \left({t}^{\left(\frac{0.3333333333333333}{2}\right)}\right)\right) - t\right)\right)\]

16. Final simplification Error: 0.3 bits

    \[\leadsto \log \left(x + y\right) + \left(\log z + \left(\left(\left(a - 0.5\right) \cdot \left(\log t \cdot 0.8333333333333334\right) + \left(a - 0.5\right) \cdot \log \left({t}^{0.16666666666666666}\right)\right) - t\right)\right)\]

Reproduce

herbie shell --seed 2020204 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (+ (log (+ x y)) (+ (- (log z) t) (* (- a 0.5) (log t))))

  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))