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

Average Error: 0.1 → 0.1

Time: 5.7s

Precision: binary64

Math

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

↓

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

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) (((double) (((double) (x + y)) + z)) - ((double) (((double) log(((double) pow(t, 0.6666666666666666)))) * z)))) - ((double) (((double) (z * ((double) (2.0 * ((double) log(((double) cbrt(((double) cbrt(t)))))))))) + ((double) (z * ((double) log(((double) cbrt(((double) cbrt(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	0.3
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-cube-cbrt 0.1

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

4. Applied log-prod 0.1

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

5. Applied distribute-lft-in 0.1

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

6. Applied associate--r+ 0.1

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

7. Simplified 0.1

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

9. Applied pow1/3 0.1

   \[\leadsto \left(\left(\left(\left(x + y\right) + z\right) - \log \left(\sqrt[3]{t} \cdot \color{blue}{{t}^{\frac{1}{3}}}\right) \cdot z\right) - z \cdot \log \left(\sqrt[3]{t}\right)\right) + \left(a - 0.5\right) \cdot b\]

10. Applied pow1/3 0.1

    \[\leadsto \left(\left(\left(\left(x + y\right) + z\right) - \log \left(\color{blue}{{t}^{\frac{1}{3}}} \cdot {t}^{\frac{1}{3}}\right) \cdot z\right) - z \cdot \log \left(\sqrt[3]{t}\right)\right) + \left(a - 0.5\right) \cdot b\]

11. Applied pow-prod-up 0.1

    \[\leadsto \left(\left(\left(\left(x + y\right) + z\right) - \log \color{blue}{\left({t}^{\left(\frac{1}{3} + \frac{1}{3}\right)}\right)} \cdot z\right) - z \cdot \log \left(\sqrt[3]{t}\right)\right) + \left(a - 0.5\right) \cdot b\]

12. Simplified 0.1

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

14. Applied add-cube-cbrt 0.1

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

15. Applied log-prod 0.1

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

16. Applied distribute-lft-in 0.1

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

17. Simplified 0.1

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

18. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020162 
(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)))