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

Average Error: 0.1 → 0.1

Time: 35.5s

Precision: binary64

Math

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

↓

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

C

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

↓

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

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

Target

Original	0.1
Target	0.4
Herbie	0.1

\[\left(\left(x + y\right) + \frac{\left(1 - {\left(\log t\right)}^{2}\right) \cdot z}{1 + \log t}\right) + \left(a - 0.5\right) \cdot b\]

Derivation

1. Initial program 0.1

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

3. Applied add-sqr-sqrt 0.1

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

4. Applied log-prod 0.1

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

5. Applied distribute-lft-in 0.1

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

6. Applied associate--r+ 0.1

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

7. Simplified 0.1

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

9. Applied add-sqr-sqrt 0.1

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

10. Applied sqrt-prod 0.1

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

11. Applied log-prod 0.1

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

12. Applied distribute-lft-in 0.1

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

13. Applied associate--r+ 0.1

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

14. Simplified 0.1

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

16. Applied sum-log 0.1

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

17. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020163 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ (+ (+ x y) (/ (* (- 1.0 (pow (log t) 2.0)) z) (+ 1.0 (log t)))) (* (- a 0.5) b))

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